 # Use a Graph to Solve an Equation

In this worksheet, students will learn how to use straight-line graphs to find solutions to linear equations. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Algebra

Curriculum subtopic:   Solving Equations and Inequalities, Algebraic Equations

Difficulty level:   ### QUESTION 1 of 10

You may already have learn about how to solve equations using algebraic methods.

However, equations can also be solved using a graphical method, which is the method we will consider in this activity.

On the graph below, we have two straight lines: y = 3x + 1 (in blue) and y = 4 (in red) We can see that these lines meet (or intersect) at the point (1, 4)

This means that when x = 1y = 4.

So x = 1 is a solution to the equation 3x + 1 = 4

We can add more horizontal lines to this graph and solve more related equations: The green line has equation y = -2 and so where it intersects the blue line we have the solution to 3x + 1 = -2.

We see that the point of intersection is (-1 , -2) and so x = -1 is the solution to this equation.

Finally, the purple line has equation y = 8.

Where it intersects the blue line, we have the solution to the equation 3x + 1 = 8.

You will notice this time that the x-value is not a whole number, so we must estimate it.

It is approximately x = 2.3 so that is our best solution.

Note: We can check our solutions using the algebraic, balancing method:

3x + 1 = 8

3x + 1 - 1 = 8 - 1

3x = 7

3x ÷ 3 = 7 ÷ 3

x = 7/3 or 2.33333...

So we can be totally sure that we have the correct answer in the last question.

In this activity, we will find solutions to pairs of equations using graphs and finding where the lines intersect.

Graphs will be shown on your screen so don't forget to use your magnifier on the toolbar if you find them tricky to see at any point.

Look at the graph below showing two lines represented graphically: Which of the equations listed below describes the relationship shown on this graph?

3x + 3 = 6

3x + 6 = 3

3x - 3 = 6

3x - 6 = 3 Use this graph to find the solution to the equation:

3x + 6 = 3

x = 1

x = 3

x = -2

x = -1

Consider the equations represented on the graph below: Use this graph to match the four equations below to their correct solutions.

## Column B

2x - 1 = 4
x = -1
2x - 1 = 2
x = 2.5
2x - 1 = -3
x = 1.5
2x - 1 = -4
x = -1.5

Consider the equations represented on the graph below: Use this graph to match the four equations below to their correct solutions.

## Column B

4x - 1 = 9
x = 1
4x - 1 = 3
x = -0.5
4x - 1 = -2
x = -1
4x - 1 = -5
x = 2.5

Consider the equations represented on the graph below: Use the graph above to solve the equation:

-3x + 4 = 1

## Column B

4x - 1 = 9
x = 1
4x - 1 = 3
x = -0.5
4x - 1 = -2
x = -1
4x - 1 = -5
x = 2.5

This graph can be used to solve the four equations listed below: Match each equation to its correct solution based on this graph.

## Column B

2 - 2x = 6
x = 1.5
2 - 2x = 3
x = -2
2 - 2x = -1
x = -0.5
2 - 2x = -3
x = 2.5

Consider the equations represented on the graph below: Use the graph above to estimate the solutions to the three equations below.

## Column B

3x + 1 = 8
x = 0.7
3x = 1 = 3
x = 2.3
3x + 1 = -3
x = -1.3

Consider the equations represented on the graph below: Use the graph above to find the approximate solution to the equation:

5x + 5 = 2

x = -0.2

x = -0.6

x = -1.4

x = -1.8

Consider the equation represented on the graph below: Use the graph above to find an approximate solution to the equation:

4 - 6x = -3

x = -0.2

x = -0.6

x = -1.4

x = -1.8

Consider the equation represented on the graph below: Use the graph above to find an approximate solution to the equation:

12 - 5x = 0

x = -0.2

x = -0.6

x = -1.4

x = -1.8

• Question 1

Look at the graph below showing two lines represented graphically: Which of the equations listed below describes the relationship shown on this graph?

3x + 6 = 3
EDDIE SAYS
The lines y = 3x + 6 and y = 3 are represented on this graph. They intersect at the point (-1, 3), so the equation which represents this relationship is: 3x + 6 = 3 Does that make sense?
• Question 2 Use this graph to find the solution to the equation:

3x + 6 = 3

x = -1
EDDIE SAYS
The point of intersection (where the two lines cross) is (-1 , 3). So the solution is to this equation is x = -1. It's as simple as that!
• Question 3

Consider the equations represented on the graph below: Use this graph to match the four equations below to their correct solutions.

## Column B

2x - 1 = 4
x = 2.5
2x - 1 = 2
x = 1.5
2x - 1 = -3
x = -1
2x - 1 = -4
x = -1.5
EDDIE SAYS
To solve these equations, we need to find the points of intersection for the relevant pairs of lines. y = 2x - 1 intersects y = 4 (blue) when x = 2.5. y = 2x - 1 intersects y = 2 (red) when x = 1.5. y = 2x - 1 intersects y = -3 (green) when x = -1. y = 2x - 1 intersects y = -4 (purple) when x = -1.5. Did you find all those matches?
• Question 4

Consider the equations represented on the graph below: Use this graph to match the four equations below to their correct solutions.

## Column B

4x - 1 = 9
x = 2.5
4x - 1 = 3
x = 1
4x - 1 = -2
x = -0.5
4x - 1 = -5
x = -1
EDDIE SAYS
Identify the points of intersection for the relevant pairs of lines. y = 4x -1 intersects y = 9 (blue) when x = 2.5. y = 4x -1 intersects y = 3 (red) when x = 1. y = 4x -1 intersects y = -2 (green) when x = -0.5. y = 4x -1 intersects y = -5 (purple) when x = -1.
• Question 5

Consider the equations represented on the graph below: Use the graph above to solve the equation:

-3x + 4 = 1

EDDIE SAYS
The line y = -3x + 4 (black) intersects the line y = 1 (green) at the point (1, 1). This means that the solution which satisfies both equations is x = 1.
• Question 6

This graph can be used to solve the four equations listed below: Match each equation to its correct solution based on this graph.

## Column B

2 - 2x = 6
x = -2
2 - 2x = 3
x = -0.5
2 - 2x = -1
x = 1.5
2 - 2x = -3
x = 2.5
EDDIE SAYS
This is just the same as our other questions, but we just need to draw the lines representing y on ourselves or mentally in our head. Then we need to find the points where these 'mental' lines cross the line y = 2 - 2x shown. When y = 6, x = -2. When y = 3, x = -0.5. When y = -1, x = 1.5. When y = -3, x = 2.5. Did you find those matches?
• Question 7

Consider the equations represented on the graph below: Use the graph above to estimate the solutions to the three equations below.

## Column B

3x + 1 = 8
x = 2.3
3x = 1 = 3
x = 0.7
3x + 1 = -3
x = -1.3
EDDIE SAYS
These lines intersect at points which do not provide exact coordinates, so we need give our best guess. The line y = 3x + 1 (black) intersects y = 8 (red) when x is approximately 2.3. The line y = 3x + 1 (black) intersects y = 3 (blue) when x is approximately 0.7. The line y = 3x + 1 intersects y = -3 (green) when x is approximately -1.3. Did you estimate those points of intersection accurately?
• Question 8

Consider the equations represented on the graph below: Use the graph above to find the approximate solution to the equation:

5x + 5 = 2

x = -0.6
EDDIE SAYS
The line y = 5x + 5 (black) intersects y = 2 (red) at approximately x = -0.6. Did you estimate that point of intersection correctly? Don't be afraid to use the toolbar to zoom in a little and take a closer look if you need to.
• Question 9

Consider the equation represented on the graph below: Use the graph above to find an approximate solution to the equation:

4 - 6x = -3

EDDIE SAYS
On the line y = 4 - 6x, when y = -3, x is approximately 1.2. Answers of either 1.2 or 1.3 are acceptable as they are both good estimates. How close were you?
• Question 10

Consider the equation represented on the graph below: Use the graph above to find an approximate solution to the equation:

12 - 5x = 0

EDDIE SAYS
The line y = 12 - 5x, intersects the line y = 0 (i.e. the x-axis), at approximately x = 2.4. Answers of either 2.4 or 2.3 are acceptable as they are both good estimates. Congratulations on completing this activity! Why not try another to really perfect your skills of solving equations using graphs?
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