## Simultaneous Equations

Simultaneous equations are systems of several equations with several variables.

The systems dealt with below are of two equations with two variables.

Solving systems with two equations means finding values of x and y that make both of the equations true.

### Linear Simultaneous Equations

There are four methods available at this level for solving simultaneous equations. The setting out for these problems needs to be done clearly and carefully, especially for the algebraic methods. Each equation should be labelled with a letter or number, and the solution set should be checked back into both equations.

1. Graphical methods. Sketch the graph of each equation. Where they intersect is the solution set.

 e.g. y = x + 1 y = -2x − 2 From the graph, the solution is the point (-1, 0)

This method is not very accurate and algebraic methods are often better used.

2. Comparison method. If the two equations have the same variable as the subject of their equations, the other sides of the equations can be equated.

e.g. y = x + 1 ............... A

y = -2x − 2 .................B

x + 1 = -2x − 2
3x = -3
x = -1

Substitute x = -1 into equation A to find y

y = -1 + 1

y = 0

The solution is ( -1, 0)

3. Substitution method. If one of the equations has a variable alone as the subject, it can be directly substituted into the other equation.

e.g. y = 3x + 4 ...................A

x + y = 12 .......................B

To find x, substitute 3x + 4 for y into equation B

x + (3x + 4) = 12
4x + 4 = 12
4x = 8
x = 2

To find y, substitute x = 2 into equation B

2 + y = 12
y = 10

The solution is (2, 10)

4. Elimination method. If the coefficients of either x or y are the same, or can easily be made the same, then either the x or the y term can be eliminated by adding or subtracting the two equations,

e.g. 4x + 3y = 24 ...................A

3x − y = 5 ..........................B

Multiply equation B by 3

9x − 3y = 15 ...................C

To find x, add equation A to equation C

4x + 3y = 24
+ 9x − 3y = 15
13x = 39
x = 3

To find y, substitute x = 3 into equation A

4 x 3 + 3y = 24
12 + 3y = 24
3y = 12
y = 4

The solution is (3, 4)

Problems Using Simultaneous Equations

Some word problems can be solved using simultaneous equations.

e.g. Five CDs and nine tapes cost \$204, while six CDs and seven tapes cost \$222.

How much would: (a) One CD cost? (b) One tape cost?

Let one CD cost \$x and one tape cost \$y

From the problem:

5x + 9y = 204 ..................A
6x + 7y = 222 ..................B

Using the elimination method:

Multiply equation A by 6 and B by 5

30x + 54y = 1224 ..................C
30x + 35y = 1110 ..................D

Subtract equation D from C to eliminate x

19y = 114
y = 6

Substitute y = 6 into equation A to find x

5x + 9 x 3 = 204
5x = 150
x = 30

All word problems should end with a word answer:

(a) The CDs cost \$30 each
(b) The tapes cost \$6 each