Unit 5 Section 5 : Simultaneous Equations

Simultaneous equations consist of two or more equations that are true at the same time. Consider the following example:

Claire and Laura are sisters; we know that

(i) Claire is the elder sister, their ages added together give 20 years, the difference between their ages is 2 years.

Let x = Claire's age, in years and y = Laura's age, in years.

 x + y = 20 x - y = 2

This is an example of a pair of simultaneous equations.

In this section we consider two methods of solving pairs of simultaneous equations like these.

Example 1

Use a graph to solve the simultaneous equations:

 x + y = 20 x - y = 2
We can rewrite the first equation to make y the subject:
 x + y = 20 y = 20 - x
For the second equation,
 x - y = 2 x = y + 2 x - 2 = y or y = x - 2
Now draw the graphs y = 20 - x and y = x - 2.
The lines cross at the point with coordinates (11, 9), so the solution of the pair of simultaneous equation is x = 11, y = 9.
Note: this means that the solution to the problem presented at the start of section 5.5 is that Claire is aged 11 and Laura is aged 9.

Example 2

Use a graph to solve the simultaneous equations:

 x + 2y = 18 3x - y = 5
First rearrange the equations in the form y = . . .
x + 2y = 18
2y = 18 - x
y =
y =
 9 -
y =
 + 9
 3x - y = 5 3x = y + 5 3x - 5 = y or y = 3x - 5

Now draw these two graphs:

The lines cross at the point with coordinates (4, 7), so the solution is x = 4, y = 7.
An alternative approach is to solve simultaneous equations algebraically, as shown in the following examples.

Example 3

Solve the simultaneous equations:

 x + 2y = 29 (1) x + y = 18 (2)

Note that the equations have been numbered (1) and (2).

Method 1 Substitution

 x + y = 18 y = 18 - x

Now replace y in equation (1) using y = 18 - x

 x + 2y = 29 x + 2(18 - x) = 29 x + 36 - 2x = 29 36 - x = 29 36 = 29 + x 36 - 29 = x x = 7

Finally, using y = 18 - x gives

 y = 18 - 7 y = 11

So the solution is x = 7, y = 11

Method 2 Elimination

Take equation (2) away from equation (1).

 x + 2y = 29 (1) x + y = 18 (2) y = 11 (1) - (2)

In equation (2), replace y with 11.

 x + 11 = 18 x = 18 - 11 x = 7

So the solution is x = 7, y = 11

Example 4

Solve the simultaneous equations:

 2x + 3y = 28 (1) x + y = 11 (2)
Method 1 Substitution

From equation (2)

 x + y = 11 y = 11 - x

Substitute this into equation (1)

 2x + 3(11 - x) = 28 2x + 33 - 3x = 28 33 - x = 28 33 = 28 + x 33 - 28 = x x = 5

Finally, use y = 11 - x

 y = 11 - x y = 11 - 5 y = 6

So the solution is x = 5, y = 6

Method 2 Elimination

Subtract 2 × equation (2) from equation (1).

 2x + 3y = 28 (1) 2x + 2y = 22 2 × (2) y = 6 (1) - 2 × (2)

Now replace y in equation (2) with 6.

 x + 6 = 11 x = 11 - 6 x = 5

So the solution is x = 5, y = 6

Example 5

Solve the simultaneous equations:

 x - 2y = 8 (1) 2x + y = 21 (2)
Method 1 Substitution

From equation (2)

 2x + y = 21 y = 21 - 2x

Substitute this into equation (1)

 x - 2y = 8 x - 2(21 - 2x) = 8 x - 42 + 4x = 8 5x - 42 = 8 5x = 8 + 42 5x = 50 x = 10

Now substitute this into y = 21 - 2x

 y = 21 - 2 × 10 y = 21 - 20 y = 1

So the solution is x = 10, y = 1

Method 2 Elimination

Subtract 2 × equation (1) from equation (2).

 2x + y = 21 (2) 2x - 4y = 16 2 × (1) 5y = 5 (2) - 2 × (1) y = 1

Now replace this in equation (1).

 x - 2y = 8 x = 8 + 2 x = 10

So the solution is x = 10, y = 1

Exercises

Question 1
(a)

Draw the lines with equations y = 10 - x and y = x + 2.

(b)

Write down the coordinates of the point where the two lines cross.

(c)

What is the solution of the pair of simultaneous equations,

 y = 10 - x y = x + 2
x = , y =
Question 2
(a)

Draw the lines with equations y = 5 - 2x and y = 4 - x.

(b)

Determine the coordinates of the point where the two lines cross.

(c)

Determine the solution of the simultaneous equations,

 2x + y = 5 x + y = 4
x = , y =
Question 3
Use a graphical method to solve the simultaneous equations,
 x - 2y = 5 x + y = 8
So the solution is x = , y =
Question 4
Use a graph to solve the simultaneous equations,
 x + 2y = 10 2x + 3y = 18
So the solution is x = , y =
Question 5
Two numbers, x and y, are such that their sum is 24 and their difference is 6.
(a)
If the numbers are x and y, write down a pair of simultaneous equations in x and y.

(b)
Use a graph to solve the simultaneous equations and hence identify the two numbers.
The two numbers are:
Question 6
Michelle obtains the solution x = 4, y = 2 to a pair of simultaneous equations by drawing the following graph:
What are the equations that she has solved?
Question 7
A pair of simultaneous equations are given below:
 2x + 4y = 14 (1) 2x + y = 8 (2)
Solve the equations.

x = , y =
Subtracting equation (2) from equation (1) helps to solve the equations because it eliminates the unknown x, leaving an equation in y only.
Question 8
Solve the following pairs of simultaneous equations, using algebraic methods:
(a)
 x + 5y = 8 x + 4y = 7
x = , y =
(b)
 2x + 3y = 16 8x + 3y = 46
x = , y =
(c)
 2x + 6y = 26 2x + 3y = 20
x = , y =
(d)
 x + 2y = 3 x + y = 7
x = , y =
(e)
 x + 3y = 18 x - 2y = 3
x = , y =
(f)
 2x + 4y = 32 2x - 3y = 11
x = , y =
Question 9
A pair of simultaneous equations is given below:
 4x + 2y = 46 (1) x + 3y = 14 (2)

Calculate the solution of this pair of equations.

x = , y =
Question 10
Solve the following pairs of simultaneous equations, using an algebraic method:
(a)
 x + 2y = 7 2x + 3y = 11
x = , y =
(b)
 4x + 9y = 47 x + 2y = 11
x = , y =
(c)
 4x + 5y = 25 x - y = 4
x = , y =
(d)
 2x + 6y = 20 x + 2y = 9
x = , y =
(e)
 x - 8y = 4 2x + y = 42
x = , y =
(f)
 4x - 2y = 24 8x - 3y = 50
x = , y =
Question 11
Look at this graph:
(a)
Show that the equation of line A is 2x + y = 8.
intercept:
equation: y = , i.e. 2x + y = 8.
(b)
Write the equation of line B.
(c)
On the graph, draw the line whose equation is y = 2x + 1.
(d)
Solve these simultaneous equations:
 y = 2x + 1 (1) 3y = 4x + 6 (2)
x = , y =

E.g. In equation (2), replace y = 2x + 1.

 3(2x + 1) = 4x + 6 6x + 3 = 4x + 6 2x = 3 x = 1.5

In equation (1), replace x = 1.5.

 y = 2 × 1.5 + 1 y = 4

So the solution is x = 1.5, y = 4

Question 12
Look at this octagon:
(a)
The line through A and H has the equation x = 10. What is the equation of the line through F and G ?
(b)
Add in the missing words to the following statement to make it correct:
x + y = 15 is the equation of the line through and .
(c)
The octagon has four lines of symmetry. One of the lines of symmetry has the equation y = x.
On the diagram, draw the line y = x.
(d)
The octagon has three other lines of symmetry.
Write the equation of one of these three other lines of symmetry.
(e)
The line through D and B has the equation 3y = x + 25.
The line through G and H has the equation x = y + 15.
Solve the simultaneous equations:
 3y = x + 25 (1) x = y + 15 (2)
x = , y =

E.g. In equation (1), replace x = y + 15.

 3y = y + 15 + 25 3y = y + 40 2y = 40 y = 20

In equation (2), replace y = 20.

 x = 20 + 15 x = 35

So the solution is x = 35, y = 20

(f)
Complete this sentence:
The line through D and B meets the line through G and H at ( , ).