Simultaneous Equations

•Elimination Method•Substitution method•Graphical Method•Matrix Method

What are they?

• Simply 2 equations– With 2 unknowns– Usually x and y

• To SOLVE the equations means we find values of x and y that– Satisfy BOTH equations [work in]– At same time [simultaneously]

Elimination Method

2x – y = 1

3x + y = 9

We have the same number of y’s in each

If we ADD the equations, the y’s disappear+

5x = 10 Divide both sides by 5

x = 2

AB

Substitute x = 2 in equation A2 x 2 – y = 1

4 – y = 1y = 3

Answerx = 2, y = 3

Elimination Method

5x + y = 17

3x + y = 11

We have the same number of y’s in each

If we SUBTRACT the equations, the y’s disappear

-

2x = 6 Divide both sides by 2

x = 3

AB

Substitute x = 3 in equation A5 x 3 + y = 17

15 + y = 17y = 2

Answerx = 3, y = 2

Elimination Method

2x + 3y = 9

2x + y = 7

We have the same number of x’s in each

If we SUBTRACT the equations,the x’s disappear

-

2y = 2 Divide both sides by 2

y = 1

AB

Substitute y = 1 in equation A2x + 3 = 9

2x = 6x = 3

Answerx = 3, y = 1

Elimination Method

4x - 3y = 14

2x + 3y = 16

We have the same number of y’s in each

If we ADD the equations,the y’s disappear

+

6x = 30 Divide both sides by 6

x = 5

AB

Substitute x = 5 in equation A20 – 3y = 14

3y = 6y = 2

Answerx = 5, y = 2

Basic steps

• Look at equations

• Same number of x’s or y’s?

• If the sign is different, ADD the equations otherwise subtract tem

• Then have ONE equation

• Solve this

• Substitute answer to get the other

• CHECK by substitution of BOTH answers

What if NOT same number of x’s or y’s?

5x + 2y = 173x + y = 10

-x = 3

In B

AB

5 x 3 + 2y = 1715 + 2y = 17

y = 1

Answerx = 3, y = 1

If we multiply A by 2 we get 2y in each

5x + 2y = 176x + 2y = 20

B

A

What if NOT same number of x’s or y’s?

3x + 6y = 214x - 2y = 8

+15x = 45

In B

AB

3 x 3 + 6y = 216y = 12

y = 2

Answerx = 3, y = 2

If we multiply A by 3 we get 6y in each

3x + 6y = 2112x - 6y = 24

B

A

x = 3

…if multiplying 1 equation doesn’t help?

3x + 7y = 265x + 2y = 24

-29y = 58

In B

AB

5x + 2 x 2 = 245x = 20

x = 4

Answerx = 4, y = 2

Multiply A by 5 & B by 3, we get 15x in each

15x + 35y = 13015x + 6y = 72B

A

y = 2

Could multiply A by 2 & Bby 7 to get 14y in each

…if multiplying 1 equation doesn’t help?

3x - 2y = 75x + 3y = 37

+19x = 95

In B

AB

5 x 5 + 3y = 373y = 12

y = 4

Answerx = 5, y = 4

Multiply A by 3 & B by 2, we get +6y & -6y

9x – 6y = 2110x + 6y = 74B

A

x = 5

Could multiply A by 5 & Bby 3 to get 15x in each

Substitution Method

Given the following equations :

y = x + 3 (i)y = 2x (ii)

Replace the y in equation (i) with 2x from equation (ii)

2x = x + 3 2x – x = 3

x = 3

Sub. x = 3 into either of the two original equations to find the value of yy = x + 3 (i)y = 3 + 3y = 6The answer is(3, 6)

A tool hire firm offers two ways in which a tool may be hired:•Plan A - $20 a day•Plan B - A payment of $40 then $10 a dayFind the number of days whereby there is no difference in the cost of hiring the tool from Plan A and Plan B.

Substitution Method

Let :y = $ in hiring toolx = no. of days hiring tool

y = 20x -----(1)y = 40 + 10x -----(2)

Sub.(1) into (2)

20x = 40 + 10x10x = 40

x = 4, y = 80

Graphical Methodx + y = 6Let x = 0 y = 6Coordinates (0, 6)Let y = 0 x = 6Coordinates (6, 0)

2x + y = 8 Let x = 0 y = 8Coordinates (0, 8) Let y = 0 2x = 8 x = 4Coordinates (4, 0)

1. x + y = 62x + y = 8

y = x + 3 y = 2x

Graphical Methody = x + 3 Let x = 0 y = 3Coordinates (0, 3)Let y = 0 x = -3Coordinates (-3, 0)

y = 2x Let x = 0 y = 0Coordinates (0, 0) Let y = 4 2x = 4 x = 2Coordinates (2, 0)