Graphs
Solution of Simultaneous Equations by Graphs

Solutions to pairs of simultaneous equations can be found by plotting lines and finding the coordinates of the point where they intersect. They can be solved analytically but this introduces a more general method which we extend in later units.

Worked Examples

1

Solve the pair of simultaneous equations

x + y = 8 and 2x + 3y = 21

Show me

First it can be helpful to write the two equations in the form y = ... .

For the first equation,

x + y = 8
y = 8 − x (subtracting x)

For the second equation,

2x + 3y = 21
3y = 21 − 2x (subtracting 2x)
y = 7 − x (dividing by 3)

Now two pairs of coordinates can be found for each line.

For y = 8 − x
If x = 0 , y = 8 so (0, 8) lies on the line
If x = 8 , y = 8 − 8 = 0 so (8, 0) lies on the line.
For y = 7 − x
If x = 0 , y = 7 − 0 = 7 so (0, 7) lies on the line.
If x = 6 , y = 7 − × 6 = 7 − 4 = 3 so (6, 3) lies on the line.

These points are then used to plot the lines shown below.

The two lines intersect at the point (3, 5), so the solution is

x = 3 and y = 5

Note

We can easily solve these equations analytically by writing

x + y = 8   ⇒  
2x + 2y = 16 1
2x + 3y = 21 2
⎫
⎬
⎭ subtract 2 from 1 to give = y = 5

Substituting back in either equation gives x; for example,

x = 8 − y = 8 − 5 = 3

Hence solution at (3, 5).

Exercises

Use the graph below to solve the simultaneous equations.

(a)

y = 8 − x

y = 2x − 1

(, )
(b)

y = 8 − x

y = 3x − 20

(, )
(c)

y = 2x − 1

y = −x − 1

(, )
(d)

3x − 20

y = −x − 1

(, )
(e)

y = −x − 14

y = 2x − 1

(, )
(f)

y = −x − 1

y = −x − 14

(, )

A discount store sells CDs and DVDs. The price of every CD is £x and the price of every DVD is £y.

(a)

Jane buys 2 CDs and 4 DVDs which cost a total of £40. Write down an equation involving x and y using this information.

=
(b)

Write down an equation in the form y = ... .

y =
(c)

Christopher buys 3 CDs and 2 DVDs which cost a total of £36. Write down a second equation using this information.

=
(d)

Write this equation in the form y = ... .

y =

Draw the graphs of both equations on the same set of axes.

(e)

What is the price of a CD?

£
(f)

What is the price of a DVD?

£

An enterprising schoolboy charges £2 to wash a car and £5 to wash and polish a car. One day he earns £80 and cleans 28 cars. Let x be the number of cars that he only washed, and y the number of cars washed and polished.

(a)

Write down two equations involving x and y.

=

=

(b)

Write these equations in the form y = ... .

y =

y =

(c)

Draw the graphs of both equations and find out how many cars were washed and polished.

The diagram shows the graphs of the equations

y = 2x + 1 and x + y = 7

Use the diagram to solve the simultaneous equations

y = 2x + 1

x + y = 7

x =

y =

The line with equation 2x + 3y = 12 is drawn on the following grid.

Solve the simultaneous equations

y = 2x − 1
2x + 3y = 12

by a graphical method.

Give your answers to 1 decimal place.

x ≈

y ≈