GCSE Maths: Equations with fractions (Foundation)
Part of the GCSE Maths linear equations chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Equations with fractions
In Chapter 3 you added and subtracted fractions by changing them to the same bottom number (the common denominator). We use the same idea to solve equations that contain fractions, but we use it to remove the fractions altogether.
Consider x + 32 = 5. The whole of the top, x + 3, is divided by 2. To undo “divide by 2” we multiply both sides by 2: x + 3 = 10, so x = 7. On the left, 2 × x + 32 = x + 3, because the 2 on the top and the 2 on the bottom cancel. Note that the top is a bracket in disguise: the fraction bar acts like a bracket around x + 3.
When there is more than one fraction, for example x2 + x3 = 10, find the lowest common multiple (LCM) of the bottom numbers. Here the LCM of 2 and 3 is 6. Multiply every term on both sides by 6:
6 × x2 = 3x 6 × x3 = 2x 6 × 10 = 60, so 3x + 2x = 60, which gives 5x = 60 and x = 12.
Why does this work? Multiplying both sides of an equation by the same number keeps it balanced. If we choose a number that every bottom divides into, each fraction turns into a whole expression. It is exactly like the way you made equivalent fractions in Chapter 3, only now the aim is to get rid of the bottoms.
The most common slips are (1) forgetting to multiply a term that is not a fraction, such as the 10 above, and (2) forgetting that a numerator with two terms needs a bracket when you multiply: 6 × x + 12 = 3(x + 1), not 3x + 1.
Key ideas
- To solve an equation with fractions, multiply every term on both sides by the lowest common multiple of the bottom numbers.
- This removes the fractions: after you multiply, the bottoms cancel.
- If a numerator has two terms, use a bracket after multiplying, then expand it.
- Multiply every term, including whole numbers. Then solve the ordinary equation that is left, and check in the original equation.
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.
- 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
- 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
- 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 - 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 - 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 - 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.
- 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
- 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
- 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 - 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 - 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.2x − 1 + 1 = 15 + 1
2x = 16 - 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 - 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 − 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.
- 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
- 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
- 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 - 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 - 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 - 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.
- 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
- 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
- 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 - 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 - 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.
Watch out
Multiply every term. In x2 + 3 = 7, multiplying by 2 gives x + 6 = 14, not x + 3 = 14. And when the top has two terms, treat it like a bracket: 6 × x + 12 = 3(x + 1).Exercise 9F Equations with fractions
Non-calculatorKey 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.
- 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
- 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
- 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 - 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 - 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 - 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.
- 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
- 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
- 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 - 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 - 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 - 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.
- 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
- 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
- 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 - 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 - 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 - 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.
- 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
- 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
- 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 - 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 - 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 - 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.
- 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
- 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
- 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 - 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 - 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 - 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.
- 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
- 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
- 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 - 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 - 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 - 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.
- 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
- 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
- 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 - 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 - 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 - 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 - 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.
- 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
- 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
- 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 - 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 - 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 - 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 - 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.
- 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
- 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
- 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 - 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 - 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 - 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 - 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.
- 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
- 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
- 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 - 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 - 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 - 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 - 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.
- 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
- 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
- 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 - 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 - 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 - 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 - 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.
- 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
- 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
- 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 - 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 - 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 - 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 - 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.
- 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
- 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
- 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 - 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 - 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 - 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.
- 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
- 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
- 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 - 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 - 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 - 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.
- 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
- 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
- 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 - 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 - 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 - 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.
- 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
- 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
- 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 - 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 - 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 - 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.
- 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
- 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
- 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 - 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 - 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.
- 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
- 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
- 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 - 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 - 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 - 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.
- 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
- 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 - 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 - 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.
- 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
- 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 - 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 - 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.
- 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
- 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 - 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 - 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.
- 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
- 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 - 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 - 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. 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. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Half of the number is x2. Adding 3 gives x2 + 3. The result is 11.
Equation: x2 + 3 = 11 - 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.x2 + 3 − 3 = 11 − 3
x2 = 8 - 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 × 8
x = 16 - Write the solution. x is now on its own, so the equation tells us its value.x = 16
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The number is 16.
Check. Half of 16 is 8, and 8 + 3 = 11 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
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. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number of pages in the book.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.A third of the pages is x3. Ana has read a third of the book and a further 10 pages, which is 40 pages altogether: x3 + 10 = 40.
Equation: x3 + 10 = 40 - Subtract 10 from both sides. This undoes the +10, because +10 − 10 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x3 + 10 − 10 = 40 − 10
x3 = 30 - 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 × 30
x = 90 - Write the solution. x is now on its own, so the equation tells us its value.x = 90
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The book has 90 pages.
Check. A third of 90 is 30, and 30 + 10 = 40 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Method. Find the term that Kai forgot to multiply, then solve correctly.
- Look at what Kai did. He multiplied both sides by 2. The x2 became x and the 7 became 14. But the + 3 on the left was left alone.Every term must be multiplied, including 3.
- Multiply every term by 2. This keeps the equation balanced and removes the fraction.2 × x2 = x 2 × 3 = 6 2 × 7 = 14
x + 6 = 14 - Solve. Subtract 6 from both sides.x = 8
Check. 82 + 3 = 4 + 3 = 7. ✓
Watch out. Multiply every term, not only the fractions.
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. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the smallest of the three numbers.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.The numbers are x, x + 4 and x + 8. The mean is the total divided by 3, so the mean is x + (x + 4) + (x + 8)3, which is 12. The top simplifies to 3x + 12.
Equation: 3x + 123 = 12 - 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
- 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 × 3x + 123 = (3x + 12) (3 goes into 3 exactly 1 time, so the bottom disappears)
3 × 12 = 36
The new equation is (3x + 12) = 36 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(3x + 12) has nothing in front of it except +1, so the brackets can simply be removed: 3x + 12
The equation is now 3x + 12 = 36 - Subtract 12 from both sides. This undoes the +12, because +12 − 12 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 12 − 12 = 36 − 12
3x = 24 - 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 = 243
x = 243 = 8 - Write the solution. x is now on its own, so the equation tells us its value.x = 8
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The numbers are 8, 12 and 16.
Check. (8 + 12 + 16) ÷ 3 = 36 ÷ 3 = 12 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
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.
- 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 = 4
- 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 × 4
x = 8 - 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: 82 = 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. 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.
- 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 + 1 = 4
- 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.x3 + 1 − 1 = 4 − 1
x3 = 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 × 3
x = 9 - Write the solution. x is now on its own, so the equation tells us its value.x = 9
Check. Substitute x = 9 into the original equation. Left side: 93 + 1 = 3 + 1 = 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. 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.
- 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 + 12 = 3
- 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
- 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 + 12 = (x + 1) (2 goes into 2 exactly 1 time, so the bottom disappears)
2 × 3 = 6
The new equation is (x + 1) = 6 - 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 = 6 - 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 = 6 − 1
x = 5 - 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 + 12 = 62 = 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.
Method. Substitute the value into both sides of the equation, work each side out, and compare.
- Write the equation and the value to test. A number is a solution of an equation if it makes both sides equal. We test it by substituting it (putting it in place of the letter).x2 + x3 = 5, x = 6
- Substitute into the left-hand side. Replace every x with the number. Use brackets around a negative number so that the signs stay clear.Left side = 62 + 63 = 3 + 2 = 5
- Substitute into the right-hand side. Do exactly the same to the other side of the equation.Right side = 5
- Compare the two sides. If the two sides are equal the number is a solution; if not, it is not.5 = 5, so yes, x = 6 is a solution.
Check. The two sides match, which is exactly what we wanted to see. ✓
Watch out. Work out the left side and the right side separately. Do not use an = sign between the two until you have shown they are equal.