Simultaneous Equations

Algebra • Topic 8 • intermediate

🔄 Simultaneous Equations

Simultaneous equations are two equations with two unknowns that must be solved together to find unique values for both variables.

Example System

2x + y = 7

x - y = 2

Find values of x and y that satisfy both equations

➕ Elimination Method

Add or subtract equations to eliminate one variable, then solve for the other.

Example: Solve 2x + y = 7 and x - y = 2

Step 1: Add the equations (to eliminate y)

(2x + y) + (x - y) = 7 + 2

3x = 9

x = 3

Step 2: Substitute x = 3 into first equation

2(3) + y = 7

6 + y = 7

y = 1

Solution: x = 3, y = 1

🔀 Substitution Method

Solve one equation for one variable, then substitute into the other equation.

Example: Solve x + 2y = 8 and 3x - y = 5

Step 1: From first equation: x = 8 - 2y

Step 2: Substitute into second equation

3(8 - 2y) - y = 5

24 - 6y - y = 5

24 - 7y = 5

-7y = -19

y = 19/7

Step 3: Find x: x = 8 - 2(19/7) = 18/7

🎯 Interactive Practice

1. Solve using elimination: x + y = 5 and x - y = 1