Simultaneous Equations: Elimination and Substitution Explained

Quick answer: Simultaneous equations are two equations with the same two unknowns, solved together. Use elimination when you can add or subtract the equations to remove one letter. Use substitution when one equation already says "y = …". Either way, find one value, put it back in to find the other, then check both equations.

Method 1: elimination

Solve 2x + y = 11 and x − y = 1.

Add the equations. The y terms cancel: 3x = 12, so x = 4.

Put x = 4 into x − y = 1: y = 3.

Check: 2(4) + 3 = 11 ✓ and 4 − 3 = 1 ✓.

When the numbers don't match

Solve 3x + 2y = 16 and 2x + 3y = 19.

Multiply the first by 3 and the second by 2 so the y terms match:

9x + 6y = 48
4x + 6y = 38

Subtract: 5x = 10, so x = 2. Then 3(2) + 2y = 16 gives y = 5.

Check: 2(2) + 3(5) = 19 ✓.

Method 2: substitution

Solve y = 2x − 1 and 3x + y = 14.

Replace y in the second equation: 3x + (2x − 1) = 14, so 5x = 15 and x = 3.

Then y = 2(3) − 1 = 5.

Method 3: graphing

Each equation is a straight line. The solution is the point where they cross. For y = 2x − 1 and 3x + y = 14, the lines meet at (3, 5). Graphing is a good check, but reading a graph can be less exact than algebra.

A word problem

A school sells adult tickets for $12 and child tickets for $7. Ten tickets bring in $95. How many of each?

Let a = adult tickets and c = child tickets.

a + c = 10
12a + 7c = 95

Substitute c = 10 − a: 12a + 70 − 7a = 95, so 5a = 25 and a = 5, c = 5.

Check: 5 × 12 + 5 × 7 = 60 + 35 = 95 ✓.

Common mistakes

Subtracting signs wrongly. Line the equations up and subtract each term carefully.

Multiplying only one side. Multiply every term in the equation.

Stopping after one value. You need both x and y.

Skipping the check. It takes ten seconds and catches most slips.

When extra support might help

Support is worth considering when algebra steps get mixed up, or before NCEA assessments.

A tutor can't guarantee a grade, but can make the method reliable. Math Make Smart offers online maths tutoring in New Zealand, and you can book a free trial lesson. For the bigger picture, see our NCEA Level 1 maths standards guide.

Frequently asked questions

What are simultaneous equations?
Two equations with the same unknowns, solved together.

When should I use elimination?
When adding or subtracting removes a letter, or can after multiplying.

When should I use substitution?
When one equation is already "y = …" or "x = …".