Concept 1

Solve systems using substitution and elimination.

Use substitution or elimination, whichever cancels faster.

Core Idea

Use substitution when one variable is already isolated or easy to isolate; use elimination when you can add or subtract the equations to cancel a variable.

Understanding

Both methods do the same thing — reduce two equations in two unknowns down to one equation in one unknown. The choice between them is about which path is shorter.

Substitution works best when one equation already looks like 𝑦 =… or 𝑥 =…. Plug that expression into the other equation, solve, then back-substitute.

Elimination works best when a variable has the same or opposite coefficient in both equations. If needed, multiply one or both equations by a constant so the coefficients match, then add or subtract to cancel that variable.

After finding one variable, always substitute back into an original equation to find the other. A quick check — plugging both values into both equations — catches arithmetic errors before you move on.

Step by Step

  1. Decide: Is a variable already isolated (→ substitution) or do coefficients align (→ elimination)?
  2. Substitution path: Solve one equation for a variable, substitute into the other, solve the resulting one-variable equation.
  3. Elimination path: Multiply equations if needed so one variable's coefficients are equal or opposite, then add/subtract.
  4. Back-substitute to find the second variable.
  5. Check both values in both original equations.

Misconceptions

  • Forgetting to distribute when substituting an expression — e.g., replacing 𝑦 in 3𝑦 with 2𝑥 +1 but writing 3 ⋅2𝑥 +1 instead of 3(2𝑥 +1).
  • Subtracting equations but making sign errors — writing 5𝑥 −( −2𝑥) as 3𝑥 instead of 7𝑥.
  • Finding 𝑥 but forgetting to solve for 𝑦, then choosing an answer that only matches 𝑥.
Question

Worked Example

What is the solution (𝑥,𝑦) to the system 3𝑥 +2𝑦 =16 and 𝑥 −2𝑦 = −8?

Select an answer to see the explanation