🔢 Systems of Equations — Elimination Method
SAT Relevance
Systems of equations appear on nearly every SAT Math section — typically 2 to 4 questions. The elimination method is often the fastest approach and is the preferred strategy of top scorers when coefficients align or can be easily manipulated.
📌 What Is the Elimination Method?
The elimination method (also called the addition/subtraction method) solves a system of two equations by adding or subtracting the equations so that one variable is eliminated, leaving a single equation in one variable.
Given a system:
The goal is to make the coefficients of one variable equal in magnitude (and opposite in sign), so that adding the equations cancels that variable.
⚡ When to Use Elimination
- ✅ Coefficients are already opposites
- ✅ Coefficients can be made opposites with simple multiplication
- ✅ The question asks for a combined expression like $3x + 2y$
- ✅ Fractions would appear with substitution
- ❌ Avoid when one variable is already isolated (use substitution instead)
📋 Step-by-Step Procedure
Step 1 Align the Equations
Write both equations in standard form: $ax + by = c$. Make sure the variables and constants line up vertically.
Step 2 Multiply (if needed)
Multiply one or both equations by constants so that the coefficients of one variable become opposites (e.g., $+3y$ and $-3y$).
Step 3 Add the Equations
Add the left sides together and the right sides together. One variable should cancel out completely.
Step 4 Solve for the Remaining Variable
You now have a single equation with one unknown. Solve it using basic algebra.
Step 5 Back-Substitute
Plug the value you found back into either original equation to find the other variable.
Step 6 Verify Your Answer
Substitute both values into the other equation to confirm. On the SAT, this catches careless errors!