GCSE Maths: Equations with fractions (Higher)

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   x + 32 = 5
2. Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.
   2  →  the lowest common multiple is 2
3. Multiply every term, on both sides, by 2. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.
   2 × x + 32 = (x + 3)   (2 goes into 2 exactly 1 time, so the bottom disappears)
   2 × 5 = 10
   The new equation is (x + 3) = 10
4. Expand the brackets. Multiply each term inside the bracket by the number outside it.
   (x + 3) has nothing in front of it except +1, so the brackets can simply be removed: x + 3
   
   The equation is now x + 3 = 10
5. Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   x + 3 − 3 = 10 − 3
   x = 7
6. Write the solution. x is now on its own, so the equation tells us its value.
   x = 7

Check. Substitute x = 7 into the original equation. Left side: 7 + 32 = 102 = 5. Right side: 5. Both sides equal 5, so the solution is correct. ✓

Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   x + 13 = 4
2. Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.
   3  →  the lowest common multiple is 3
3. Multiply every term, on both sides, by 3. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.
   3 × x + 13 = (x + 1)   (3 goes into 3 exactly 1 time, so the bottom disappears)
   3 × 4 = 12
   The new equation is (x + 1) = 12
4. Expand the brackets. Multiply each term inside the bracket by the number outside it.
   (x + 1) has nothing in front of it except +1, so the brackets can simply be removed: x + 1
   
   The equation is now x + 1 = 12
5. Subtract 1 from both sides. This undoes the +1, because +1 − 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   x + 1 − 1 = 12 − 1
   x = 11
6. Write the solution. x is now on its own, so the equation tells us its value.
   x = 11

Check. Substitute x = 11 into the original equation. Left side: 11 + 13 = 123 = 4. Right side: 4. Both sides equal 4, so the solution is correct. ✓

Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   x − 25 = 2
2. Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.
   5  →  the lowest common multiple is 5
3. Multiply every term, on both sides, by 5. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.
   5 × x − 25 = (x − 2)   (5 goes into 5 exactly 1 time, so the bottom disappears)
   5 × 2 = 10
   The new equation is (x − 2) = 10
4. Expand the brackets. Multiply each term inside the bracket by the number outside it.
   (x − 2) has nothing in front of it except +1, so the brackets can simply be removed: x − 2
   
   The equation is now x − 2 = 10
5. Add 2 to both sides. This undoes the −2, because −2 + 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   x − 2 + 2 = 10 + 2
   x = 12
6. Write the solution. x is now on its own, so the equation tells us its value.
   x = 12

Check. Substitute x = 12 into the original equation. Left side: 12 − 25 = 105 = 2. Right side: 2. Both sides equal 2, so the solution is correct. ✓

Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   x + 74 = 3
2. Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.
   4  →  the lowest common multiple is 4
3. Multiply every term, on both sides, by 4. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.
   4 × x + 74 = (x + 7)   (4 goes into 4 exactly 1 time, so the bottom disappears)
   4 × 3 = 12
   The new equation is (x + 7) = 12
4. Expand the brackets. Multiply each term inside the bracket by the number outside it.
   (x + 7) has nothing in front of it except +1, so the brackets can simply be removed: x + 7
   
   The equation is now x + 7 = 12
5. Subtract 7 from both sides. This undoes the +7, because +7 − 7 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   x + 7 − 7 = 12 − 7
   x = 5
6. Write the solution. x is now on its own, so the equation tells us its value.
   x = 5

Check. Substitute x = 5 into the original equation. Left side: 5 + 74 = 124 = 3. Right side: 3. Both sides equal 3, so the solution is correct. ✓

Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   x − 42 = 6
2. Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.
   2  →  the lowest common multiple is 2
3. Multiply every term, on both sides, by 2. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.
   2 × x − 42 = (x − 4)   (2 goes into 2 exactly 1 time, so the bottom disappears)
   2 × 6 = 12
   The new equation is (x − 4) = 12
4. Expand the brackets. Multiply each term inside the bracket by the number outside it.
   (x − 4) has nothing in front of it except +1, so the brackets can simply be removed: x − 4
   
   The equation is now x − 4 = 12
5. Add 4 to both sides. This undoes the −4, because −4 + 4 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   x − 4 + 4 = 12 + 4
   x = 16
6. Write the solution. x is now on its own, so the equation tells us its value.
   x = 16

Check. Substitute x = 16 into the original equation. Left side: 16 − 42 = 122 = 6. Right side: 6. Both sides equal 6, so the solution is correct. ✓

Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   x + 56 = 2
2. Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.
   6  →  the lowest common multiple is 6
3. Multiply every term, on both sides, by 6. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.
   6 × x + 56 = (x + 5)   (6 goes into 6 exactly 1 time, so the bottom disappears)
   6 × 2 = 12
   The new equation is (x + 5) = 12
4. Expand the brackets. Multiply each term inside the bracket by the number outside it.
   (x + 5) has nothing in front of it except +1, so the brackets can simply be removed: x + 5
   
   The equation is now x + 5 = 12
5. Subtract 5 from both sides. This undoes the +5, because +5 − 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   x + 5 − 5 = 12 − 5
   x = 7
6. Write the solution. x is now on its own, so the equation tells us its value.
   x = 7

Check. Substitute x = 7 into the original equation. Left side: 7 + 56 = 126 = 2. Right side: 2. Both sides equal 2, so the solution is correct. ✓

Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   2x + 13 = 5
2. Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.
   3  →  the lowest common multiple is 3
3. Multiply every term, on both sides, by 3. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.
   3 × 2x + 13 = (2x + 1)   (3 goes into 3 exactly 1 time, so the bottom disappears)
   3 × 5 = 15
   The new equation is (2x + 1) = 15
4. Expand the brackets. Multiply each term inside the bracket by the number outside it.
   (2x + 1) has nothing in front of it except +1, so the brackets can simply be removed: 2x + 1
   
   The equation is now 2x + 1 = 15
5. Subtract 1 from both sides. This undoes the +1, because +1 − 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   2x + 1 − 1 = 15 − 1
   2x = 14
6. Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.
   2x2 = 142
   x = 142 = 7
7. Write the solution. x is now on its own, so the equation tells us its value.
   x = 7

Check. Substitute x = 7 into the original equation. Left side: 2 × 7 + 13 = 153 = 5. Right side: 5. Both sides equal 5, so the solution is correct. ✓

Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   3x − 24 = 4
2. Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.
   4  →  the lowest common multiple is 4
3. Multiply every term, on both sides, by 4. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.
   4 × 3x − 24 = (3x − 2)   (4 goes into 4 exactly 1 time, so the bottom disappears)
   4 × 4 = 16
   The new equation is (3x − 2) = 16
4. Expand the brackets. Multiply each term inside the bracket by the number outside it.
   (3x − 2) has nothing in front of it except +1, so the brackets can simply be removed: 3x − 2
   
   The equation is now 3x − 2 = 16
5. Add 2 to both sides. This undoes the −2, because −2 + 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   3x − 2 + 2 = 16 + 2
   3x = 18
6. Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.
   3x3 = 183
   x = 183 = 6
7. Write the solution. x is now on its own, so the equation tells us its value.
   x = 6

Check. Substitute x = 6 into the original equation. Left side: 3 × 6 − 24 = 164 = 4. Right side: 4. Both sides equal 4, so the solution is correct. ✓

Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   5x + 32 = 14
2. Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.
   2  →  the lowest common multiple is 2
3. Multiply every term, on both sides, by 2. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.
   2 × 5x + 32 = (5x + 3)   (2 goes into 2 exactly 1 time, so the bottom disappears)
   2 × 14 = 28
   The new equation is (5x + 3) = 28
4. Expand the brackets. Multiply each term inside the bracket by the number outside it.
   (5x + 3) has nothing in front of it except +1, so the brackets can simply be removed: 5x + 3
   
   The equation is now 5x + 3 = 28
5. Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   5x + 3 − 3 = 28 − 3
   5x = 25
6. Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.
   5x5 = 255
   x = 255 = 5
7. Write the solution. x is now on its own, so the equation tells us its value.
   x = 5

Check. Substitute x = 5 into the original equation. Left side: 5 × 5 + 32 = 282 = 14. Right side: 14. Both sides equal 14, so the solution is correct. ✓

Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   4x − 13 = 5
2. Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.
   3  →  the lowest common multiple is 3
3. Multiply every term, on both sides, by 3. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.
   3 × 4x − 13 = (4x − 1)   (3 goes into 3 exactly 1 time, so the bottom disappears)
   3 × 5 = 15
   The new equation is (4x − 1) = 15
4. Expand the brackets. Multiply each term inside the bracket by the number outside it.
   (4x − 1) has nothing in front of it except +1, so the brackets can simply be removed: 4x − 1
   
   The equation is now 4x − 1 = 15
5. Add 1 to both sides. This undoes the −1, because −1 + 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   4x − 1 + 1 = 15 + 1
   4x = 16
6. Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.
   4x4 = 164
   x = 164 = 4
7. Write the solution. x is now on its own, so the equation tells us its value.
   x = 4

Check. Substitute x = 4 into the original equation. Left side: 4 × 4 − 13 = 153 = 5. Right side: 5. Both sides equal 5, so the solution is correct. ✓

Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   2x + 57 = 3
2. Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.
   7  →  the lowest common multiple is 7
3. Multiply every term, on both sides, by 7. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.
   7 × 2x + 57 = (2x + 5)   (7 goes into 7 exactly 1 time, so the bottom disappears)
   7 × 3 = 21
   The new equation is (2x + 5) = 21
4. Expand the brackets. Multiply each term inside the bracket by the number outside it.
   (2x + 5) has nothing in front of it except +1, so the brackets can simply be removed: 2x + 5
   
   The equation is now 2x + 5 = 21
5. Subtract 5 from both sides. This undoes the +5, because +5 − 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   2x + 5 − 5 = 21 − 5
   2x = 16
6. Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.
   2x2 = 162
   x = 162 = 8
7. Write the solution. x is now on its own, so the equation tells us its value.
   x = 8

Check. Substitute x = 8 into the original equation. Left side: 2 × 8 + 57 = 217 = 3. Right side: 3. Both sides equal 3, so the solution is correct. ✓

Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   3x + 15 = 2
2. Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.
   5  →  the lowest common multiple is 5
3. Multiply every term, on both sides, by 5. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.
   5 × 3x + 15 = (3x + 1)   (5 goes into 5 exactly 1 time, so the bottom disappears)
   5 × 2 = 10
   The new equation is (3x + 1) = 10
4. Expand the brackets. Multiply each term inside the bracket by the number outside it.
   (3x + 1) has nothing in front of it except +1, so the brackets can simply be removed: 3x + 1
   
   The equation is now 3x + 1 = 10
5. Subtract 1 from both sides. This undoes the +1, because +1 − 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   3x + 1 − 1 = 10 − 1
   3x = 9
6. Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.
   3x3 = 93
   x = 93 = 3
7. Write the solution. x is now on its own, so the equation tells us its value.
   x = 3

Check. Substitute x = 3 into the original equation. Left side: 3 × 3 + 15 = 105 = 2. Right side: 2. Both sides equal 2, so the solution is correct. ✓

Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   x2 + x3 = 10
2. Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.
   2, 3  →  the lowest common multiple is 6
3. Multiply every term, on both sides, by 6. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.
   6 × x2 = 3x   (2 goes into 6 exactly 3 times, so the bottom disappears)
   6 × x3 = 2x   (3 goes into 6 exactly 2 times, so the bottom disappears)
   6 × 10 = 60
   The new equation is 3x + 2x = 60
4. Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.
   On the left side: the x terms 3x + 2x make 5x.
   The equation is now 5x = 60
5. Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.
   5x5 = 605
   x = 605 = 12
6. Write the solution. x is now on its own, so the equation tells us its value.
   x = 12

Check. Substitute x = 12 into the original equation. Left side: 122 + 123 = 6 + 4 = 10. Right side: 10. Both sides equal 10, so the solution is correct. ✓

Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   x3 + x4 = 7
2. Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.
   3, 4  →  the lowest common multiple is 12
3. Multiply every term, on both sides, by 12. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.
   12 × x3 = 4x   (3 goes into 12 exactly 4 times, so the bottom disappears)
   12 × x4 = 3x   (4 goes into 12 exactly 3 times, so the bottom disappears)
   12 × 7 = 84
   The new equation is 4x + 3x = 84
4. Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.
   On the left side: the x terms 4x + 3x make 7x.
   The equation is now 7x = 84
5. Divide both sides by 7. 7x means 7 multiplied by x. The inverse of multiplying by 7 is dividing by 7, and we do it to both sides to keep the balance. This leaves x on its own.
   7x7 = 847
   x = 847 = 12
6. Write the solution. x is now on its own, so the equation tells us its value.
   x = 12

Check. Substitute x = 12 into the original equation. Left side: 123 + 124 = 4 + 3 = 7. Right side: 7. Both sides equal 7, so the solution is correct. ✓

Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   x2 − x5 = 6
2. Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.
   2, 5  →  the lowest common multiple is 10
3. Multiply every term, on both sides, by 10. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.
   10 × x2 = 5x   (2 goes into 10 exactly 5 times, so the bottom disappears)
   10 × (−x5) = −2x   (5 goes into 10 exactly 2 times, so the bottom disappears)
   10 × 6 = 60
   The new equation is 5x − 2x = 60
4. Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.
   On the left side: the x terms 5x − 2x make 3x.
   The equation is now 3x = 60
5. Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.
   3x3 = 603
   x = 603 = 20
6. Write the solution. x is now on its own, so the equation tells us its value.
   x = 20

Check. Substitute x = 20 into the original equation. Left side: 202 − 205 = 10 − 4 = 6. Right side: 6. Both sides equal 6, so the solution is correct. ✓

Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   x4 + x2 = 9
2. Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.
   4, 2  →  the lowest common multiple is 4
3. Multiply every term, on both sides, by 4. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.
   4 × x4 = x   (4 goes into 4 exactly 1 time, so the bottom disappears)
   4 × x2 = 2x   (2 goes into 4 exactly 2 times, so the bottom disappears)
   4 × 9 = 36
   The new equation is x + 2x = 36
4. Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.
   On the left side: the x terms x + 2x make 3x.
   The equation is now 3x = 36
5. Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.
   3x3 = 363
   x = 363 = 12
6. Write the solution. x is now on its own, so the equation tells us its value.
   x = 12

Check. Substitute x = 12 into the original equation. Left side: 124 + 122 = 3 + 6 = 9. Right side: 9. Both sides equal 9, so the solution is correct. ✓

Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   x3 + 2 = x2
2. Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.
   3, 2  →  the lowest common multiple is 6
3. Multiply every term, on both sides, by 6. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.
   6 × x3 = 2x   (3 goes into 6 exactly 2 times, so the bottom disappears)
   6 × 2 = 12
   6 × x2 = 3x   (2 goes into 6 exactly 3 times, so the bottom disappears)
   The new equation is 2x + 12 = 3x
4. Remove the x term from the left side: subtract 2x from both sides. We want all the x terms on one side. The left side has 2x; subtracting 2x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.
   2x + 12 − 2x = 3x − 2x
   12 = x
5. Write the solution. x is now on its own, so the equation tells us its value.
   x = 12

Check. Substitute x = 12 into the original equation. Left side: 123 + 2 = 4 + 2 = 6. Right side: 122 = 6. Both sides equal 6, so the solution is correct. ✓

Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   x5 + 3 = x2
2. Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.
   5, 2  →  the lowest common multiple is 10
3. Multiply every term, on both sides, by 10. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.
   10 × x5 = 2x   (5 goes into 10 exactly 2 times, so the bottom disappears)
   10 × 3 = 30
   10 × x2 = 5x   (2 goes into 10 exactly 5 times, so the bottom disappears)
   The new equation is 2x + 30 = 5x
4. Remove the x term from the left side: subtract 2x from both sides. We want all the x terms on one side. The left side has 2x; subtracting 2x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.
   2x + 30 − 2x = 5x − 2x
   30 = 3x
5. Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.
   303 = 3x3
   x = 303 = 10
6. Write the solution. x is now on its own, so the equation tells us its value.
   x = 10

Check. Substitute x = 10 into the original equation. Left side: 105 + 3 = 2 + 3 = 5. Right side: 102 = 5. Both sides equal 5, so the solution is correct. ✓

Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   x4 + 3 = 8
2. Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   x4 + 3 − 3 = 8 − 3
   x4 = 5
3. Multiply both sides by 4. x4 means x divided by 4. The inverse of dividing by 4 is multiplying by 4. We do it to both sides to keep the balance.
   4 × x4 = 4 × 5
   x = 20
4. Write the solution. x is now on its own, so the equation tells us its value.
   x = 20

Check. Substitute x = 20 into the original equation. Left side: 204 + 3 = 5 + 3 = 8. Right side: 8. Both sides equal 8, so the solution is correct. ✓

Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   x3 − 5 = 1
2. Add 5 to both sides. This undoes the −5, because −5 + 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   x3 − 5 + 5 = 1 + 5
   x3 = 6
3. Multiply both sides by 3. x3 means x divided by 3. The inverse of dividing by 3 is multiplying by 3. We do it to both sides to keep the balance.
   3 × x3 = 3 × 6
   x = 18
4. Write the solution. x is now on its own, so the equation tells us its value.
   x = 18

Check. Substitute x = 18 into the original equation. Left side: 183 − 5 = 6 − 5 = 1. Right side: 1. Both sides equal 1, so the solution is correct. ✓

Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   x6 + 2 = 7
2. Subtract 2 from both sides. This undoes the +2, because +2 − 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   x6 + 2 − 2 = 7 − 2
   x6 = 5
3. Multiply both sides by 6. x6 means x divided by 6. The inverse of dividing by 6 is multiplying by 6. We do it to both sides to keep the balance.
   6 × x6 = 6 × 5
   x = 30
4. Write the solution. x is now on its own, so the equation tells us its value.
   x = 30

Check. Substitute x = 30 into the original equation. Left side: 306 + 2 = 5 + 2 = 7. Right side: 7. Both sides equal 7, so the solution is correct. ✓

Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   x2 − 3 = 4
2. Add 3 to both sides. This undoes the −3, because −3 + 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   x2 − 3 + 3 = 4 + 3
   x2 = 7
3. Multiply both sides by 2. x2 means x divided by 2. The inverse of dividing by 2 is multiplying by 2. We do it to both sides to keep the balance.
   2 × x2 = 2 × 7
   x = 14
4. Write the solution. x is now on its own, so the equation tells us its value.
   x = 14

Check. Substitute x = 14 into the original equation. Left side: 142 − 3 = 7 − 3 = 4. Right side: 4. Both sides equal 4, so the solution is correct. ✓

Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.

Key words. To solve an equation means to find the value of x that makes it true. Whatever you do to one side you must do to the other side, so that it stays balanced, like a pair of scales.

Method. Solve an equation containing fractions: (1) find the common denominator, (2) multiply every term by it to remove the fractions, (3) solve the simple equation that is left.

1. Write the equation. An equation says two things are equal. We want the value of x that makes both sides equal.
   x + 12 + x + 23 = 7
2. Find the common denominator of all the bottoms. Fractions are awkward to solve with, so we will remove them. To do that we multiply by something that every bottom divides into: the lowest common multiple of the bottoms.
   2, 3
   Common denominator = 6
3. Multiply every term, on both sides, by the common denominator. Doing the same thing to both sides keeps the equation balanced. Multiply every single term, including any plain numbers. This removes all the fractions.
   6 × x + 12 = 3(x + 1)   (the bottom 2 goes into 6 exactly 3 times, so the bottom disappears and the top is multiplied by 3)
   6 × x + 23 = 2(x + 2)   (the bottom 3 goes into 6 exactly 2 times, so the bottom disappears and the top is multiplied by 2)
   6 × 7 = 42   (a plain number or term with no bottom is simply multiplied by 6)
4. Write the new equation. There are no fractions left. Check that every term from the original equation has been multiplied.
   3(x + 1) + 2(x + 2) = 42
5. Expand the brackets. Multiply each term inside by the number or letter outside. A minus sign in front of a bracket changes the sign of every term inside it.
   Expand 3(x + 1):
   3 × x = 3x
   3 × 1 = 3
   so it is 3x + 3
   
   Expand 2(x + 2):
   2 × x = 2x
   2 × 2 = 4
   so it is 2x + 4
6. Collect like terms on each side. Add or subtract the x terms together, and the plain numbers together, on each side separately.
   3x + 3 + 2x + 4 = 42
   5x + 7 = 42
7. Get the x terms on one side and the plain numbers on the other. Use inverse operations on both sides. The inverse of adding is subtracting and the inverse of subtracting is adding. Always do the same to both sides.
   Start: 5x + 7 = 42
   Now get the numbers on the other side. Subtract 7 from both sides: 5x = 35
8. Divide both sides by the number in front of x (which is 5). 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5. This leaves x on its own.
   x = 35 ÷ 5 = 7

Check. Put x = 7 back into the original equation. Left side: 8/2 + 9/3 = 7. Right side: 7 = 7. Both sides are equal, so the solution is correct. ✓

Watch out. Multiply every term, including the numbers that are not fractions. And a solution is only allowed if it does not make a bottom equal to 0.

Key words. To solve an equation means to find the value of x that makes it true. Whatever you do to one side you must do to the other side, so that it stays balanced, like a pair of scales.

Method. Solve an equation containing fractions: (1) find the common denominator, (2) multiply every term by it to remove the fractions, (3) solve the simple equation that is left.

1. Write the equation. An equation says two things are equal. We want the value of x that makes both sides equal.
   x + 12 − x − 24 = 4
2. Find the common denominator of all the bottoms. Fractions are awkward to solve with, so we will remove them. To do that we multiply by something that every bottom divides into: the lowest common multiple of the bottoms.
   2, 4
   Common denominator = 4
3. Multiply every term, on both sides, by the common denominator. Doing the same thing to both sides keeps the equation balanced. Multiply every single term, including any plain numbers. This removes all the fractions.
   4 × x + 12 = 2(x + 1)   (the bottom 2 goes into 4 exactly 2 times, so the bottom disappears and the top is multiplied by 2)
   4 × x − 24 = (x − 2)   (the bottom 4 goes into 4 exactly 1 time, so the bottom disappears and the top is multiplied by 1)
   4 × 4 = 16   (a plain number or term with no bottom is simply multiplied by 4)
4. Write the new equation. There are no fractions left. Check that every term from the original equation has been multiplied.
   2(x + 1) − (x − 2) = 16
5. Expand the brackets. Multiply each term inside by the number or letter outside. A minus sign in front of a bracket changes the sign of every term inside it.
   Expand 2(x + 1):
   2 × x = 2x
   2 × 1 = 2
   so it is 2x + 2
6. Collect like terms on each side. Add or subtract the x terms together, and the plain numbers together, on each side separately.
   2x + 2 − x + 2 = 16
   x + 4 = 16
7. Get the x terms on one side and the plain numbers on the other. Use inverse operations on both sides. The inverse of adding is subtracting and the inverse of subtracting is adding. Always do the same to both sides.
   Start: x + 4 = 16
   Now get the numbers on the other side. Subtract 4 from both sides: x = 12
8. Read off the answer. x is already on its own.
   x = 12

Check. Put x = 12 back into the original equation. Left side: 13/2 − 10/4 = 4. Right side: 4 = 4. Both sides are equal, so the solution is correct. ✓

Watch out. Multiply every term, including the numbers that are not fractions. And a solution is only allowed if it does not make a bottom equal to 0.

Key words. To solve an equation means to find the value of x that makes it true. Whatever you do to one side you must do to the other side, so that it stays balanced, like a pair of scales.

Method. Solve an equation containing fractions: (1) find the common denominator, (2) multiply every term by it to remove the fractions, (3) solve the simple equation that is left.

1. Write the equation. An equation says two things are equal. We want the value of x that makes both sides equal.
   3x − 22 − 3x − 33 = 3
2. Find the common denominator of all the bottoms. Fractions are awkward to solve with, so we will remove them. To do that we multiply by something that every bottom divides into: the lowest common multiple of the bottoms.
   2, 3
   Common denominator = 6
3. Multiply every term, on both sides, by the common denominator. Doing the same thing to both sides keeps the equation balanced. Multiply every single term, including any plain numbers. This removes all the fractions.
   6 × 3x − 22 = 3(3x − 2)   (the bottom 2 goes into 6 exactly 3 times, so the bottom disappears and the top is multiplied by 3)
   6 × 3x − 33 = 2(3x − 3)   (the bottom 3 goes into 6 exactly 2 times, so the bottom disappears and the top is multiplied by 2)
   6 × 3 = 18   (a plain number or term with no bottom is simply multiplied by 6)
4. Write the new equation. There are no fractions left. Check that every term from the original equation has been multiplied.
   3(3x − 2) − 2(3x − 3) = 18
5. Expand the brackets. Multiply each term inside by the number or letter outside. A minus sign in front of a bracket changes the sign of every term inside it.
   Expand 3(3x − 2):
   3 × 3x = 9x
   3 × (−2) = −6
   so it is 9x − 6
   
   Expand 2(3x − 3):
   2 × 3x = 6x
   2 × (−3) = −6
   so it is 6x − 6
6. Collect like terms on each side. Add or subtract the x terms together, and the plain numbers together, on each side separately.
   9x − 6 − 6x + 6 = 18
   3x = 18
7. Get the x terms on one side and the plain numbers on the other. Use inverse operations on both sides. The inverse of adding is subtracting and the inverse of subtracting is adding. Always do the same to both sides.
   Start: 3x = 18
8. Divide both sides by the number in front of x (which is 3). 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3. This leaves x on its own.
   x = 18 ÷ 3 = 6

Check. Put x = 6 back into the original equation. Left side: 16/2 − 15/3 = 3. Right side: 3 = 3. Both sides are equal, so the solution is correct. ✓

Watch out. Multiply every term, including the numbers that are not fractions. And a solution is only allowed if it does not make a bottom equal to 0.

Key words. To solve an equation means to find the value of x that makes it true. Whatever you do to one side you must do to the other side, so that it stays balanced, like a pair of scales.

Method. Solve an equation containing fractions: (1) find the common denominator, (2) multiply every term by it to remove the fractions, (3) solve the simple equation that is left.

1. Write the equation. An equation says two things are equal. We want the value of x that makes both sides equal.
   5x − 12 + 2x − 24 = 8
2. Find the common denominator of all the bottoms. Fractions are awkward to solve with, so we will remove them. To do that we multiply by something that every bottom divides into: the lowest common multiple of the bottoms.
   2, 4
   Common denominator = 4
3. Multiply every term, on both sides, by the common denominator. Doing the same thing to both sides keeps the equation balanced. Multiply every single term, including any plain numbers. This removes all the fractions.
   4 × 5x − 12 = 2(5x − 1)   (the bottom 2 goes into 4 exactly 2 times, so the bottom disappears and the top is multiplied by 2)
   4 × 2x − 24 = (2x − 2)   (the bottom 4 goes into 4 exactly 1 time, so the bottom disappears and the top is multiplied by 1)
   4 × 8 = 32   (a plain number or term with no bottom is simply multiplied by 4)
4. Write the new equation. There are no fractions left. Check that every term from the original equation has been multiplied.
   2(5x − 1) + (2x − 2) = 32
5. Expand the brackets. Multiply each term inside by the number or letter outside. A minus sign in front of a bracket changes the sign of every term inside it.
   Expand 2(5x − 1):
   2 × 5x = 10x
   2 × (−1) = −2
   so it is 10x − 2
6. Collect like terms on each side. Add or subtract the x terms together, and the plain numbers together, on each side separately.
   10x − 2 + 2x − 2 = 32
   12x − 4 = 32
7. Get the x terms on one side and the plain numbers on the other. Use inverse operations on both sides. The inverse of adding is subtracting and the inverse of subtracting is adding. Always do the same to both sides.
   Start: 12x − 4 = 32
   Now get the numbers on the other side. Add 4 to both sides: 12x = 36
8. Divide both sides by the number in front of x (which is 12). 12x means 12 multiplied by x. The inverse of multiplying by 12 is dividing by 12. This leaves x on its own.
   x = 36 ÷ 12 = 3

Check. Put x = 3 back into the original equation. Left side: 14/2 + 4/4 = 8. Right side: 8 = 8. Both sides are equal, so the solution is correct. ✓

Watch out. Multiply every term, including the numbers that are not fractions. And a solution is only allowed if it does not make a bottom equal to 0.

Key words. To solve an equation means to find the value of x that makes it true. Whatever you do to one side you must do to the other side, so that it stays balanced, like a pair of scales.

Method. Solve an equation containing fractions: (1) find the common denominator, (2) multiply every term by it to remove the fractions, (3) solve the simple equation that is left.

1. Write the equation. An equation says two things are equal. We want the value of x that makes both sides equal.
   2x + 24 = 3x − 15
2. Find the common denominator of all the bottoms. Fractions are awkward to solve with, so we will remove them. To do that we multiply by something that every bottom divides into: the lowest common multiple of the bottoms.
   4, 5
   Common denominator = 20
3. Multiply every term, on both sides, by the common denominator. Doing the same thing to both sides keeps the equation balanced. Multiply every single term, including any plain numbers. This removes all the fractions.
   20 × 2x + 24 = 5(2x + 2)   (the bottom 4 goes into 20 exactly 5 times, so the bottom disappears and the top is multiplied by 5)
   20 × 3x − 15 = 4(3x − 1)   (the bottom 5 goes into 20 exactly 4 times, so the bottom disappears and the top is multiplied by 4)
4. Write the new equation. There are no fractions left. Check that every term from the original equation has been multiplied.
   5(2x + 2) = 4(3x − 1)
5. Expand the brackets. Multiply each term inside by the number or letter outside. A minus sign in front of a bracket changes the sign of every term inside it.
   Expand 5(2x + 2):
   5 × 2x = 10x
   5 × 2 = 10
   so it is 10x + 10
   
   Expand 4(3x − 1):
   4 × 3x = 12x
   4 × (−1) = −4
   so it is 12x − 4
6. Collect like terms on each side. Add or subtract the x terms together, and the plain numbers together, on each side separately.
   10x + 10 = 12x − 4
   10x + 10 = 12x − 4
7. Get the x terms on one side and the plain numbers on the other. Use inverse operations on both sides. The inverse of adding is subtracting and the inverse of subtracting is adding. Always do the same to both sides.
   Start: 10x + 10 = 12x − 4
   Subtract 10x from both sides (we move the smaller x term, to keep the numbers positive): 10 = 2x − 4
   Now add 4 to both sides: 14 = 2x
8. Divide both sides by the number in front of x (which is 2). 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2. This leaves x on its own.
   x = 14 ÷ 2 = 7

Check. Put x = 7 back into the original equation. Left side: 16/4 = 4. Right side: 20/5 = 4. Both sides are equal, so the solution is correct. ✓

Watch out. Multiply every term, including the numbers that are not fractions. And a solution is only allowed if it does not make a bottom equal to 0.

Key words. To solve an equation means to find the value of x that makes it true. Whatever you do to one side you must do to the other side, so that it stays balanced, like a pair of scales.

Method. Solve an equation containing fractions: (1) find the common denominator, (2) multiply every term by it to remove the fractions, (3) solve the simple equation that is left.

1. Write the equation. An equation says two things are equal. We want the value of x that makes both sides equal.
   3x + 14 = 2x + 23
2. Find the common denominator of all the bottoms. Fractions are awkward to solve with, so we will remove them. To do that we multiply by something that every bottom divides into: the lowest common multiple of the bottoms.
   4, 3
   Common denominator = 12
3. Multiply every term, on both sides, by the common denominator. Doing the same thing to both sides keeps the equation balanced. Multiply every single term, including any plain numbers. This removes all the fractions.
   12 × 3x + 14 = 3(3x + 1)   (the bottom 4 goes into 12 exactly 3 times, so the bottom disappears and the top is multiplied by 3)
   12 × 2x + 23 = 4(2x + 2)   (the bottom 3 goes into 12 exactly 4 times, so the bottom disappears and the top is multiplied by 4)
4. Write the new equation. There are no fractions left. Check that every term from the original equation has been multiplied.
   3(3x + 1) = 4(2x + 2)
5. Expand the brackets. Multiply each term inside by the number or letter outside. A minus sign in front of a bracket changes the sign of every term inside it.
   Expand 3(3x + 1):
   3 × 3x = 9x
   3 × 1 = 3
   so it is 9x + 3
   
   Expand 4(2x + 2):
   4 × 2x = 8x
   4 × 2 = 8
   so it is 8x + 8
6. Collect like terms on each side. Add or subtract the x terms together, and the plain numbers together, on each side separately.
   9x + 3 = 8x + 8
   9x + 3 = 8x + 8
7. Get the x terms on one side and the plain numbers on the other. Use inverse operations on both sides. The inverse of adding is subtracting and the inverse of subtracting is adding. Always do the same to both sides.
   Start: 9x + 3 = 8x + 8
   The x terms should be on one side. Subtract 8x from both sides: x + 3 = 8
   Now get the numbers on the other side. Subtract 3 from both sides: x = 5
8. Read off the answer. x is already on its own.
   x = 5

Check. Put x = 5 back into the original equation. Left side: 16/4 = 4. Right side: 12/3 = 4. Both sides are equal, so the solution is correct. ✓

Watch out. Multiply every term, including the numbers that are not fractions. And a solution is only allowed if it does not make a bottom equal to 0.

Key words. To solve an equation means to find the value of x that makes it true. Whatever you do to one side you must do to the other side, so that it stays balanced, like a pair of scales.

Method. Solve an equation containing fractions: (1) find the common denominator, (2) multiply every term by it to remove the fractions, (3) solve the simple equation that is left.

1. Write the equation. An equation says two things are equal. We want the value of x that makes both sides equal.
   2x − 32 = 3x + 54
2. Find the common denominator of all the bottoms. Fractions are awkward to solve with, so we will remove them. To do that we multiply by something that every bottom divides into: the lowest common multiple of the bottoms.
   2, 4
   Common denominator = 4
3. Multiply every term, on both sides, by the common denominator. Doing the same thing to both sides keeps the equation balanced. Multiply every single term, including any plain numbers. This removes all the fractions.
   4 × 2x − 32 = 2(2x − 3)   (the bottom 2 goes into 4 exactly 2 times, so the bottom disappears and the top is multiplied by 2)
   4 × 3x + 54 = (3x + 5)   (the bottom 4 goes into 4 exactly 1 time, so the bottom disappears and the top is multiplied by 1)
4. Write the new equation. There are no fractions left. Check that every term from the original equation has been multiplied.
   2(2x − 3) = (3x + 5)
5. Expand the brackets. Multiply each term inside by the number or letter outside. A minus sign in front of a bracket changes the sign of every term inside it.
   Expand 2(2x − 3):
   2 × 2x = 4x
   2 × (−3) = −6
   so it is 4x − 6
6. Collect like terms on each side. Add or subtract the x terms together, and the plain numbers together, on each side separately.
   4x − 6 = 3x + 5
   4x − 6 = 3x + 5
7. Get the x terms on one side and the plain numbers on the other. Use inverse operations on both sides. The inverse of adding is subtracting and the inverse of subtracting is adding. Always do the same to both sides.
   Start: 4x − 6 = 3x + 5
   The x terms should be on one side. Subtract 3x from both sides: x − 6 = 5
   Now get the numbers on the other side. Add 6 to both sides: x = 11
8. Read off the answer. x is already on its own.
   x = 11

Check. Put x = 11 back into the original equation. Left side: 19/2 = 19/2. Right side: 38/4 = 19/2. Both sides are equal, so the solution is correct. ✓

Watch out. Multiply every term, including the numbers that are not fractions. And a solution is only allowed if it does not make a bottom equal to 0.

Key words. The denominator is the bottom of a fraction. We cannot divide by zero, so a value of x that makes a denominator zero is never a solution.

Method. Multiply both sides by the bottom of the fraction to remove it, then solve the ordinary equation that is left.

1. Write down the equation. The unknown is in the bottom (denominator) of a fraction. The bottom can never be zero, so we will check this at the end.
   12x = 3
2. Multiply both sides by x. This removes the fraction. Multiplying by the bottom cancels it, because anything divided by itself is 1. We do it to both sides to keep the balance.
   12x × x = 3 × x
   12 = 3x
3. Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.
   123 = 3x3
   x = 123 = 4
4. Write the solution. x is now on its own, so the equation tells us its value.
   x = 4
5. Check that the bottom is not zero. If a solution made the bottom equal to 0 the fraction would be impossible, so it would have to be rejected.
   x = 4 when x = 4, which is not 0, so the solution is allowed.

Check. Substitute x = 4: the bottom is 4, so the fraction is 124 = 3, which equals 3. ✓

Watch out. Multiply the whole of the other side by the bottom, so the right-hand side becomes a bracket. Never solve by "flipping" only part of it.

Key words. The denominator is the bottom of a fraction. We cannot divide by zero, so a value of x that makes a denominator zero is never a solution.

Method. Multiply both sides by the bottom of the fraction to remove it, then solve the ordinary equation that is left.

1. Write down the equation. The unknown is in the bottom (denominator) of a fraction. The bottom can never be zero, so we will check this at the end.
   6x − 1 = 2
2. Multiply both sides by (x − 1). This removes the fraction. Multiplying by the bottom cancels it, because anything divided by itself is 1. We do it to both sides to keep the balance.
   6x − 1 × (x − 1) = 2 × (x − 1)
   6 = 2(x − 1)
3. Expand the brackets. Multiply each term inside the bracket by the number outside it.
   Expand 2(x − 1):
   2 × x = 2x
   2 × (−1) = −2
   
   The equation is now 6 = 2x − 2
4. Add 2 to both sides. This undoes the −2, because −2 + 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   6 + 2 = 2x − 2 + 2
   8 = 2x
5. Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.
   82 = 2x2
   x = 82 = 4
6. Write the solution. x is now on its own, so the equation tells us its value.
   x = 4
7. Check that the bottom is not zero. If a solution made the bottom equal to 0 the fraction would be impossible, so it would have to be rejected.
   x − 1 = 3 when x = 4, which is not 0, so the solution is allowed.

Check. Substitute x = 4: the bottom is 3, so the fraction is 63 = 2, which equals 2. ✓

Watch out. Multiply the whole of the other side by the bottom, so the right-hand side becomes a bracket. Never solve by "flipping" only part of it.

Key words. The denominator is the bottom of a fraction. We cannot divide by zero, so a value of x that makes a denominator zero is never a solution.

Method. Multiply both sides by the bottom of the fraction to remove it, then solve the ordinary equation that is left.

1. Write down the equation. The unknown is in the bottom (denominator) of a fraction. The bottom can never be zero, so we will check this at the end.
   182x − 1 = 6
2. Multiply both sides by (2x − 1). This removes the fraction. Multiplying by the bottom cancels it, because anything divided by itself is 1. We do it to both sides to keep the balance.
   182x − 1 × (2x − 1) = 6 × (2x − 1)
   18 = 6(2x − 1)
3. Expand the brackets. Multiply each term inside the bracket by the number outside it.
   Expand 6(2x − 1):
   6 × 2x = 12x
   6 × (−1) = −6
   
   The equation is now 18 = 12x − 6
4. Add 6 to both sides. This undoes the −6, because −6 + 6 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   18 + 6 = 12x − 6 + 6
   24 = 12x
5. Divide both sides by 12. 12x means 12 multiplied by x. The inverse of multiplying by 12 is dividing by 12, and we do it to both sides to keep the balance. This leaves x on its own.
   2412 = 12x12
   x = 2412 = 2
6. Write the solution. x is now on its own, so the equation tells us its value.
   x = 2
7. Check that the bottom is not zero. If a solution made the bottom equal to 0 the fraction would be impossible, so it would have to be rejected.
   2x − 1 = 3 when x = 2, which is not 0, so the solution is allowed.

Check. Substitute x = 2: the bottom is 3, so the fraction is 183 = 6, which equals 6. ✓

Watch out. Multiply the whole of the other side by the bottom, so the right-hand side becomes a bracket. Never solve by "flipping" only part of it.

Key words. The denominator is the bottom of a fraction. We cannot divide by zero, so a value of x that makes a denominator zero is never a solution.

Method. Multiply both sides by the bottom of the fraction to remove it, then solve the ordinary equation that is left.

1. Write down the equation. The unknown is in the bottom (denominator) of a fraction. The bottom can never be zero, so we will check this at the end.
   15x + 2 = 3
2. Multiply both sides by (x + 2). This removes the fraction. Multiplying by the bottom cancels it, because anything divided by itself is 1. We do it to both sides to keep the balance.
   15x + 2 × (x + 2) = 3 × (x + 2)
   15 = 3(x + 2)
3. Expand the brackets. Multiply each term inside the bracket by the number outside it.
   Expand 3(x + 2):
   3 × x = 3x
   3 × 2 = 6
   
   The equation is now 15 = 3x + 6
4. Subtract 6 from both sides. This undoes the +6, because +6 − 6 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   15 − 6 = 3x + 6 − 6
   9 = 3x
5. Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.
   93 = 3x3
   x = 93 = 3
6. Write the solution. x is now on its own, so the equation tells us its value.
   x = 3
7. Check that the bottom is not zero. If a solution made the bottom equal to 0 the fraction would be impossible, so it would have to be rejected.
   x + 2 = 5 when x = 3, which is not 0, so the solution is allowed.

Check. Substitute x = 3: the bottom is 5, so the fraction is 155 = 3, which equals 3. ✓

Watch out. Multiply the whole of the other side by the bottom, so the right-hand side becomes a bracket. Never solve by "flipping" only part of it.