Systems of equations are one of the highest-frequency topics on the SAT Math section. You will see 2-4 questions involving systems on every test.

The good news? There are only 3 methods to learn, and each one works best in specific situations.

Method 1: Substitution

Use when: One variable is already isolated (like ).

The steps:

  1. Solve one equation for one variable
  2. Substitute into the other equation
  3. Solve, then back-substitute
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Example 1: Substitution in Action

Solve the system:


Step 1: is already isolated in the first equation.

Step 2: Substitute for in the second equation:

Step 3: Simplify and solve:



Back-substitute:

Solution:

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Method 2: Elimination

Use when: Both equations are in standard form () and you can easily cancel a variable.

The steps:

  1. Multiply one or both equations so a variable has opposite coefficients
  2. Add the equations to eliminate that variable
  3. Solve for the remaining variable
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Example 2: Elimination in Action

Solve the system:


The terms are already opposites ( and ). Add the equations:


Substitute back: , so , .

Solution:

Method 3: Graphing

Use when: The question asks for the number of solutions or gives you a graph.

Two lines can intersect in exactly one point (one solution), be parallel (no solution), or be the same line (infinitely many solutions).

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Common Mistakes to Avoid

  • Forgetting to substitute back: Finding is only half the job. Always find too (unless the question only asks for ).
  • Sign errors in elimination: When multiplying an equation by , flip ALL signs, not just the first term.
  • Using the wrong method: If one variable is already isolated, substitution is faster. If both equations are in standard form with matching coefficients, elimination wins.
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SAT Pro Tip

When the SAT asks for the value of an expression like or (not individual values), you can often use elimination to find the expression directly without solving for and separately. This saves 30+ seconds.

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