Text
Unit F3 Section 1
Simultaneous Linear Equations

A pair of equations which use both terms at the same time, such as

x + 2y = 8
2x + y = 7

are known as a pair of simultaneous equations. It is a straightforward process to manipulate these equations into one linear equation by eliminating one of the unknowns (x or y). The equation can then be readily solved.

Worked Examples

1

Solve the pair of simultaneous equations

x + 2y = 8
2x + y = 7

First it is helpful to label the equations (1) and (2).

x + 2y = 8 (1)
2x + y = 7 (2)

Equation (1) is multiplied by 2, so that it contains the same number of x's as equation (2).

Let the new equation be labelled (3).

2x + 4y = 16 (3)   ( 2 × (1) )
2x + y = 7 (2)

Equation (2) is now subtracted from equation (3).

2x + 4y = 16 (3)
2x + y = 7 (2)
3y = 9 (3) − (2)

Solving 3y = 9 gives y = 3.

This value of y can now be substituted into equation (1) to give:

x + 2 × 3 = 8
x + 6 = 8

Solving this gives x = 2. So the solution to the equation is x = 2, y = 3.
2

Solve the simultaneous equations

3x + 5y = 2
−4x + 7y = −30

First label the equations (1) and (2) as shown below.

3x + 5y = 2 (1)
−4x + 7y = −30 (2)

Then multiply equation (1) by 4 and equation (2) by 3 to make the number of x's in both equations the same.

12x + 20y = 8 (3)   ( 4 × (1) )
−12x + 21y = −90 (4)   ( 3 × (2) )

Now add together equations (3) and (4) to give

12x + 20y = 8 (3)
−12x + 21y = −90 (4)
41y = −82 (3) + (4)

Solving the equation 41y = −82 gives y = −2.

This value for y can be substituted into equation (1) to give

3x + 5 × (−2) = 2
or 3x − 10 = 2

Solving this equation gives:

3x − 10 = 2
3x = 12
x =
= 4

So the solution is x = 4 and y = −2.

Note

It is a good idea to check that solutions are correct by substituting these values back into the original equations. Here,

3 × 4 + 5 × (−2) = 2
and
−4 × 4 + 7 × (−2) = −30

You must check both equations to make sure that you have the correct answer.

3

Denise sells 300 tickets for a concert. Some tickets are sold to adults at £5 each and some are sold to children at £4 each. If she collects in £1444 in ticket sales, how many tickets have been sold to adults and how many to children?

Let x = number of adults' tickets

and y = number of children's tickets.

She has sold 300 tickets, so

x + y = 300

The value of the adult tickets sold is £5x, and the value of the children's tickets is £4y.

As the value of all the tickets sold is £1444, then

5x + 4y = 1444

The two simultaneous equations

x + y = 300 (1)
5x + 4y = 1444 (2)

can now be solved. First multiply equation (1) by 5 and subtract equation (2) to give

5x + 5y = 1500 (3)   ( 5 × (1) )
5x + 4y = 1444 (2)
y = 56 (3) – (2)

This value can then be substituted into equation (1) to give

x + 56 = 300
or x = 244

So the solution is x = 244 and y = 56. That is, 244 adults' tickets and 56 children's tickets have been sold.

Investigation

Consider the following simultaneous equations.

2x + y = 6 (1)
x = 1 − y (2)

If (2) is substituted for x into (1), then

2(1 − y) + y = 6
2 − y + y = 6
2 = 6

Find out where the problem lies.

Exercises

Solve each pair of simultaneous equations.

(a)
x + 2y = 5
3x + y = 5

x =

y =

(b)
3x + 2y = 19
x + 5y = 15

x =

y =

(c)
x − 2y = 4
4x + 3y = 49

x =

y =

(d)
2x + 3y = 14
5x + 2y = 24

x =

y =

(e)
3x + 4y = 2
7x − 5y = 9

x =

y =

(f)
4x + 2y = 16
−3x + 2y = −19

x =

y =

(g)
5x + y = 2
−4x + 3y = 44

x =

y =

(h)
6x − 4y = 12
−9x + 2y = −66

x =

y =

(i)
7x − 2y = 23
3x + 4y = 39

x =

y =

(j)
8x + 4y = 7
−12x + 8y = −6

x =

y =

(k)
4x − 2y = −0.1
5x + 2y = 1.5

x =

y =

(l)
6x − 5y = 41
4x + 15y = 31

x =

y =

(m)
−2x + 5y = 14
10x + 7y = 26

x =

y =

(n)
8x + 5y = −29
3x − 7y = −2

x =

y =

(o)
6x − 5y = −14
18x − 4y = 6

x =

y =

(p)
6x − 8y = −2
5x + 2y = 1.8

x =

y =

(q)
x − y = 0
x + y = 10

x =

y =

(r)
x − y = −1
x + y = 10

x =

y =

Find the coordinates of the point of intersection of the lines:

(a)

x + y = 8   and   y = 2x − 1

(,)
(b)

x + y = 10   and   y = 2x + 1

(,)
(c)

x + y = 4   and   y = 2 −

,

Dexter rows a boat in a river where there is a steady current. He travels at 1.3 mph upstream and 3.7 mph downstream.

Use a pair of simultaneous equations to find v, the speed of the boat in still water, and c, the speed of the current.

v =

c =

A machine sells tickets for travel on a tram system. A single ticket costs £1 and a return ticket costs £2. In one day, the machine sells 100 tickets and takes £172. How many of each type of ticket were sold?

return tickets and single tickets were sold.

An American tourist takes 200 notes to a bank. They are a mixture of $50 and $100 notes. The tourist claims that there is $13 950 in total.

(a)

Find out how many of each type of note there should be.

$100 notes and $50 notes
(b)

In fact the value of the notes turns out to be $13 900. How many of each type of note does this mean there should be? What error do you think the tourist made?

$100 notes and $50 notes

A group of people boarded a bus at a bus station. The fare to the first bus stop was £3 and the fare to the second bus stop was £5. The driver collected £238 in fares. When the bus left the first bus stop, there were 38 people on the bus. How many people got off the bus at the first stop?

Solve the simultaneous equations,

2a + 4c = 13
a + 3c = 8

a =

c =