GCSE Maths: Solving equations using graphs, and interpreting solutions (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)
Solving equations using graphs, and interpreting solutions
Equations can also be solved with a picture. The straight line y = 2x + 1 shows, for every value of x, the value of 2x + 1 (you will study these lines in Chapter 12). Suppose we want to solve 2x + 1 = 9. That means: for which x does the height of the line equal 9?
Draw a horizontal line at y = 9. It meets the sloping line at one point. Go straight down to the x-axis from that point and read the value: x = 4. This is the solution.
A graph gives approximate answers, because we read a scale by eye. Algebra gives exact answers. But graphs are useful when an equation is hard to solve by hand, and they help you see why an equation has the solution it does.
Two sloping lines. To solve an equation with x on both sides, such as 3x − 2 = x + 4, draw a line for each side: y = 3x − 2 and y = x + 4. The solution is the x-coordinate of the point where the two lines cross, because that is the one place where both sides have the same value.
Does the answer make sense? A solution of an equation is a number, but in a story the number must also fit the situation. You cannot buy 8.3 tickets, or be −4 years old, or have a rectangle with a negative length. After solving a problem from words, always ask: “Is this a sensible answer for the question?” If not, either you have made a mistake, or the situation is impossible, and you should say so.
Key ideas
- To solve f(x) = number from a graph: go across from the number on the y-axis to the line, then down to the x-axis.
- For an equation with x on both sides, plot a line for each side. The x-coordinate of the crossing point is the solution.
- Graph answers are approximate; algebra gives exact answers.
- Always check that a solution makes sense in the context (whole number? positive?).
Method. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 2x + 1, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals 9.2x + 1 = 9 means y = 9
- Find 9 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = 9.Start at y = 9 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = 4.
Check. Check by substituting: 2 × 4 + 1 = 9. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
Method. Draw (or read) the graph of each side of the equation. The x-coordinate of the point where the lines cross is the solution.
- Think of each side of the equation as a straight line. The left-hand side is the line y = 3x − 2 and the right-hand side is the line y = x + 4. The equation is true where the two lines have the same height.y = 3x − 2 and y = x + 4
- Find where the two lines cross. At the crossing point both lines have the same x and the same y, so that x makes the equation true.The lines cross at (3, 7).
- Read off the x-coordinate. The question asks for x, so we use the first number of the point.x = 3
Check. Left side: 3 × 3 − 2 = 7. Right side: 1 × 3 + 4 = 7. Equal. ✓
Watch out. The answer is the x-coordinate only, not the whole point.
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 tickets Ben bought.
- 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.Each ticket costs £6, so x tickets cost 6x pounds. Ben spent £50.
Equation: 6x = 50 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 506
x = 506 = 253 - Write the solution. x is now on its own, so the equation tells us its value.x = 253
- 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.x = 253 = 8.333… This is not a whole number, and Ben can only buy a whole number of tickets. So Ben cannot have spent exactly £50 on tickets. (He could buy 8 tickets for £48.)
Check. Substitute x = 253 into the original equation. Left side: 6 × 253 = 50. Right side: 50. Both sides equal 50, so the solution is correct. ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Watch out
The solution is the x-value (how far across), not the y-value (how far up). When you draw the line y = 9, the 9 is the number on the y-axis that you start from. The answer you read from the x-axis is 4.Exercise 9I Graphs and interpreting solutions
Non-calculatorMethod. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 2x + 1, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals 7.2x + 1 = 7 means y = 7
- Find 7 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = 7.Start at y = 7 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = 3.
Check. Check by substituting: 2 × 3 + 1 = 7. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
Method. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 2x + 1, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals 11.2x + 1 = 11 means y = 11
- Find 11 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = 11.Start at y = 11 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = 5.
Check. Check by substituting: 2 × 5 + 1 = 11. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
Method. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 2x + 1, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals 3.2x + 1 = 3 means y = 3
- Find 3 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = 3.Start at y = 3 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = 1.
Check. Check by substituting: 2 × 1 + 1 = 3. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
Method. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 2x + 1, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals 5.2x + 1 = 5 means y = 5
- Find 5 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = 5.Start at y = 5 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = 2.
Check. Check by substituting: 2 × 2 + 1 = 5. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
Method. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 2x + 1, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals 9.2x + 1 = 9 means y = 9
- Find 9 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = 9.Start at y = 9 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = 4.
Check. Check by substituting: 2 × 4 + 1 = 9. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
Method. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 2x + 1, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals 8.2x + 1 = 8 means y = 8
- Find 8 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = 8.Start at y = 8 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = 3.5.
Check. Check by substituting: 2 × 72 + 1 = 8. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
Method. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 3x − 2, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals 7.3x − 2 = 7 means y = 7
- Find 7 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = 7.Start at y = 7 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = 3.
Check. Check by substituting: 3 × 3 − 2 = 7. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
Method. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 3x − 2, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals 10.3x − 2 = 10 means y = 10
- Find 10 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = 10.Start at y = 10 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = 4.
Check. Check by substituting: 3 × 4 − 2 = 10. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
Method. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 3x − 2, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals 1.3x − 2 = 1 means y = 1
- Find 1 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = 1.Start at y = 1 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = 1.
Check. Check by substituting: 3 × 1 − 2 = 1. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
Method. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 3x − 2, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals -2.3x − 2 = -2 means y = -2
- Find -2 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = -2.Start at y = -2 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = 0.
Check. Check by substituting: 3 × 0 − 2 = −2. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
Method. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 3x − 2, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals 4.3x − 2 = 4 means y = 4
- Find 4 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = 4.Start at y = 4 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = 2.
Check. Check by substituting: 3 × 2 − 2 = 4. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
Method. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 3x − 2, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals -5.3x − 2 = -5 means y = -5
- Find -5 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = -5.Start at y = -5 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = −1.
Check. Check by substituting: 3 × (−1) − 2 = −5. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
Method. Draw (or read) the graph of each side of the equation. The x-coordinate of the point where the lines cross is the solution.
- Think of each side of the equation as a straight line. The left-hand side is the line y = 3x − 2 and the right-hand side is the line y = x + 4. The equation is true where the two lines have the same height.y = 3x − 2 and y = x + 4
- Find where the two lines cross. At the crossing point both lines have the same x and the same y, so that x makes the equation true.The lines cross at (3, 7).
- Read off the x-coordinate. The question asks for x, so we use the first number of the point.x = 3
Check. Left side: 3 × 3 − 2 = 7. Right side: 1 × 3 + 4 = 7. Equal. ✓
Watch out. The answer is the x-coordinate only, not the whole point.
Method. Draw (or read) the graph of each side of the equation. The x-coordinate of the point where the lines cross is the solution.
- Think of each side of the equation as a straight line. The left-hand side is the line y = 2x + 1 and the right-hand side is the line y = −x + 10. The equation is true where the two lines have the same height.y = 2x + 1 and y = −x + 10
- Find where the two lines cross. At the crossing point both lines have the same x and the same y, so that x makes the equation true.The lines cross at (3, 7).
- Read off the x-coordinate. The question asks for x, so we use the first number of the point.x = 3
Check. Left side: 2 × 3 + 1 = 7. Right side: −1 × 3 + 10 = 7. Equal. ✓
Watch out. The answer is the x-coordinate only, not the whole point.
Method. Read the crossing point from the graph, then confirm with algebra.
- Read the graph. The solution is the x-coordinate of the point where the lines cross.The lines cross at (4, 7), so x = 4.
- Solve by algebra. Subtract x from both sides, then add 1 to both sides.2x − 1 = x + 3
2x − x − 1 = 3
x − 1 = 3
x = 4 - Compare. The graph and the algebra agree.x = 4 both ways.
Check. 2 × 4 − 1 = 7 and 4 + 3 = 7. ✓
Watch out. The graph gives only an approximate reading. Algebra gives the exact value.
Method. Check her answer by substitution and find which axis she read.
- Test Ava's answer. Put x = 9 into 2x + 1.2 × 9 + 1 = 19, but we need 9. So x = 9 is wrong.
- Find the mistake. The number 9 is where we start on the y-axis. The solution is where the lines meet, read on the x-axis.The point where the lines meet is (4, 9). The x-coordinate is 4.
- Give the correct solution. x = 4
Check. 2 × 4 + 1 = 9. ✓
Watch out. Read the value that the question asks for: x comes from the horizontal axis.
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 tickets.
- 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.Each ticket costs £6, so x tickets cost 6x pounds, and the total is 50.
Equation: 6x = 50 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 506
x = 506 = 253 - Write the solution. x is now on its own, so the equation tells us its value.x = 253
- 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 solution is x = 253 = 8.33…, which is not a whole number of tickets. No, Ben cannot be right: he can only buy a whole number of tickets.
Check. 6 × 8 = 48 and 6 × 9 = 54, so £50 is between two possible totals ✓
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 = Dee's 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.Multiply by 4 (4x) and add 3 (4x + 3). The result is 20.
Equation: 4x + 3 = 20 - 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.4x + 3 − 3 = 20 − 3
4x = 17 - 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 = 174
x = 174 - Write the solution. x is now on its own, so the equation tells us its value.x = 174
- 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 solution is x = 174 = 4.25, which is not a whole number. No: Dee is not telling the truth, because her number would have to be a whole number.
Check. 4 × 4 + 3 = 19 and 4 × 5 + 3 = 23, so 20 is not possible ✓
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 extra gigabytes.
- 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 monthly charge is 15 and the extra cost is 4x. The total is 51.
Equation: 4x + 15 = 51 - Subtract 15 from both sides. This undoes the +15, because +15 − 15 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x + 15 − 15 = 51 − 15
4x = 36 - 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 = 364
x = 364 = 9 - Write the solution. x is now on its own, so the equation tells us its value.x = 9
- 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.x = 9 gigabytes, which is a positive whole number. This is a sensible answer.
Check. 4 × 9 = 36 and 36 + 15 = 51 ✓
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 cups of flour.
- 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 equation is given: 3x + 8 = 2.
Equation: 3x + 8 = 2 - Subtract 8 from both sides. This undoes the +8, because +8 − 8 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 8 − 8 = 2 − 8
3x = −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 = −63
x = −63 = −2 - Write the solution. x is now on its own, so the equation tells us its value.x = −2
- 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.x = −2. A recipe cannot use a negative number of cups of flour, so there is a mistake in the story or the equation.
Check. 3 × (−2) + 8 = 2 ✓
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. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 2x + 1, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals 8.2x + 1 = 8 means y = 8
- Find 8 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = 8.Start at y = 8 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = 3.5.
Check. Check by substituting: 2 × 72 + 1 = 8. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
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.2x + 1 = 7
- 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 = 7 − 1
2x = 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 = 62
x = 62 = 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: 2 × 3 + 1 = 6 + 1 = 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 ÷.