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13.3 Simultaneous equations 1

13.3 Simultaneous equations 1

!e sum of two numbers is 83. !e di%erence between the two numbers is 18. What are the numbers?

Call the numbers x and y. !en x + y = 83
and x − y = 18 Simultaneous equations
are both true at

Now you have two equations and two unknowns. !ese are simultaneous the same time, or
equations.
simultaneously, for the
You need to 'nd values of x and y that solve both equations simultaneously. two variables.

Rewrite the second equation: x = 18 + y

Substitute this into the 'rst equation: 18 + y + y = 83
A 2y = 83 − 18 = 65
A y = 32.5
!en substitute this value to 'nd x: x = 18 + y = 18 + 32.5 = 50.5

Check: x + y = 50.5 + 32.5 = 83 and x − y = 50.5 − 32.5 = 18.

Worked example 13.3

Solve these simultaneous equations. 3x + 2y = 60
y = 2x − 5

Substitute the second equation into the fi rst.
3x + 2(2x − 5) = 60 Put 2x − 5 inside a pair of brackets.
3x + 4x − 10 = 60 Multiply out the brackets.
7x = 70 Add 3x and 4x; add 10 to both sides.
x = 10 Divide by 7.
Substitute this into the second equation.
y = 2 × 10 − 5 = 15 Check these values in the fi rst equation: (3 × 10) + (2 × 15) = 60

) Exercise 13.3

1 Solve these simultaneous equations. y = 2x − 1 y = x + 4 Start with 2x − 1 = x + 4.

2 Solve these simultaneous equations. y = x − 9 y = 3x + 1
3 Solve these simultaneous equations. y = 9 − 2x y = x − 12

4 Solve each pair of simultaneous equations.
a x + y = 1 b x + y = 19 c x + y = −2
y = 2x − 8 y = 5x + 1 y = x − 10

5 Solve these simultaneous equations. 3x = y x = y − 16
6 Solve these simultaneous equations. y = 2x x = 2y − 9

7 Solve these simultaneous equations. y = 3(x + 5) 2x + y = 0
8 Solve these simultaneous equations. 2x + 5y = 22 y = x − 4

128 13 Equations and inequalities