Year 9AC9M10A02

Simultaneous Equations

Extension — this is the Year 10 descriptor: solve linear inequalities and simultaneous linear equations in 2 variables; interpret solutions graphically.

Two equations, two unknowns. Get rid of one letter and you are left with something you can already solve.

Worked example

e.g.

2x + y = 7 \quad \text{and} \quad 3x - y = 8

Line the equations up and number them $2x + y = 7 \ \ (1) \quad 3x - y = 8 \ \ (2)$
Look for a letter with equal and opposite coefficients $+y \text{ and } -y$
Add the equations to eliminate that letter $(1)+(2): \ 5x = 15$
Solve for the remaining letter $x = 3$
Substitute back into either original equation $2(3) + y = 7$
Solve for the second letter $y = 1$
Check in the OTHER equation $3(3) - 1 = 8 \ \checkmark$

Practice

Solve for $x$ and $y$ :

x + y = 10 \quad x - y = 4

Answer

x = 7,\ y = 3

2x + y = 11 \quad x - y = 1

Answer

x = 4,\ y = 3

3x + 2y = 16 \quad x - 2y = 0

Answer

x = 4,\ y = 2

x + 3y = 14 \quad 2x - 3y = 1

Answer

x = 5,\ y = 3

4x + y = 14 \quad 2x + y = 8

Answer

x = 3,\ y = 2

5x - 2y = 8 \quad 3x + 2y = 16

Answer

x = 3,\ y = 3.5

2x + 3y = 13 \quad 4x - 3y = 5

Answer

x = 3,\ y = \tfrac{7}{3}

y = 2x + 1 \quad y = x + 4

Answer

x = 3,\ y = 7

3x + y = 10 \quad x + y = 6

Answer

x = 2,\ y = 4

7x - 3y = 1 \quad 2x + 3y = 17

Answer

x = 2,\ y = \tfrac{13}{3}