Solving Systems of Equations by Elimination – Easy and Hard Examples

Struggling to teach systems of equations at home?

The elimination method is a powerful technique that often makes solving systems easier—especially when substitution gets messy.

Let’s walk through two examples to show how it works.

This is covered in Jacobs Algebra chapter 7, and in Geometry in chapter 12 Algebra review, and in chapter 2.2 of Algebra 2 with Trig. 

✅ Simple Example

System:

1. 3x + 2y = 16
2. 5x – 2y = 4

Add both equations:

→ 8x = 20
→ x = 2.5

Now plug x into one equation:

3(2.5) + 2y = 16 → 7.5 + 2y = 16 → y = 4.25

Solution: (2.5, 4.25)

🤔 What if it’s not that easy?

System:

1. 2x + 3y = 7
2. 5x – 2y = 8

We’ll eliminate y by scaling the equations:

• Multiply first by 2 → 4x + 6y = 14
• Multiply second by 3 → 15x – 6y = 24

Add:

→ 19x = 38
→ x = 2

Then plug back in:

2(2) + 3y = 7 → y = 1

Solution: (2, 1)

🧠 When to Use Elimination:

• When one variable has matching (or easily matchable) coefficients
• When substitution introduces fractions early
• When you’re solving under time pressure (like on the ACT)

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