Simultaneous Equations

When you have two unknowns, you need two equations to find them. Imagine you buy 2 pens and 3 notebooks for NPR 150, and 3 pens and 1 notebook for NPR 100. How much does each item cost? Simultaneous equations will give you the answer.

Method 1: Substitution

Express one variable in terms of the other, then substitute.

Worked Example: Solve x + y = 10 and 2x - y = 5

Step 1: From the first equation, y = 10 - x

Step 2: Substitute into the second: 2x - (10 - x) = 5

Step 3: Simplify: 3x = 15, so x = 5

Step 4: y = 10 - 5 = 5. Solution: x = 5, y = 5

Method 2: Elimination

Add or subtract the equations to eliminate one variable.

Worked Example: Solve 3x + 2y = 12 and 5x - 2y = 4

The y-terms cancel when we add: 8x = 16, so x = 2

Substitute back: 3(2) + 2y = 12, so y = 3. Solution: x = 2, y = 3

When coefficients do not match, multiply one or both equations first.

Example: Solve 2x + 3y = 7 and 3x + 2y = 8

Multiply first by 3 and second by 2: 6x + 9y = 21 and 6x + 4y = 16

Subtract: 5y = 5, so y = 1. Then x = 2.

Method 3: Graphical Method

Each equation represents a straight line. The intersection point gives the solution.

• Lines intersect at one point: one unique solution
• Parallel lines: no solution
• Overlapping lines: infinitely many solutions

Tip: Substitution works best when one variable is easy to isolate. Elimination works best when coefficients can be matched easily.

Key Takeaways

• Use substitution when one variable is easy to isolate
• Use elimination when coefficients can be matched easily
• Always verify your answer by plugging back into both original equations
• Graphically, the solution is where the two lines cross