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,
(ii) their ages added together give 20 years,
(iii) 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

Start with equation (2)

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 = 222 × (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 = 162 × (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.

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(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.

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(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
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So the solution is x = , y =
Question 4
Use a graph to solve the simultaneous equations,
x + 2y = 10
2x + 3y = 18
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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.
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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:
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(a)
Show that the equation of line A is 2x + y = 8.
gradient:
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:
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(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 ( , ).