Systems of equations are one of the most-tested topics on the SAT. You'll need to find values that satisfy two equations at once. The two main approaches are substitution and elimination — and knowing when to use each one saves you serious time.

Substitution Method

Use substitution when one variable is already isolated (or easy to isolate).

Steps:

  1. Solve one equation for one variable
  2. Plug that expression into the other equation
  3. Solve, then back-substitute

Example 1:


Substitute the first equation into the second:




Back-substitute:

Solution:

Elimination Method

Use elimination when coefficients are easy to match up.

Steps:

  1. Line up the equations vertically
  2. Multiply one or both equations so a variable cancels
  3. Add the equations together

Example 2:


The -terms already cancel when we add:


Plug back in:

Solution:

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When the SAT Asks for an Expression

Sometimes the SAT doesn't ask for or individually — it asks for something like or . In these cases, elimination can get you the answer directly.

Example 3: Given and , what is ?

Subtract the second from the first:

Done — no need to find and separately.

No Solution and Infinite Solutions

The SAT tests whether you recognize special cases:

Example 4: For what value of does this system have no solution?


Multiply the second equation by 2:

For no solution, the left sides must match but the right sides must differ: , so .

Practice Problems

Problem 1: Solve using substitution: and

Solution




,

Problem 2: Solve using elimination: and

Solution

Subtract: , so
Plug in:

Problem 3: For what value of does and have infinitely many solutions?

Solution

Multiply the second by 3: . For infinite solutions: , so .

Key Takeaways

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