How to Solve Systems of Equations by Substitution (Step by Step)

To solve a system of equations by substitution, isolate one variable in one equation, plug that expression into the other equation, solve for the remaining variable, and then back-substitute to get the second value. The result is an ordered pair (x, y) that satisfies both equations.

Below you'll find the method spelled out, two fully worked examples, and the mistakes students make most often.

A fully worked example

Solve the system:

• 2x + y = 5
• x - y = 1

Step 1 — Isolate a variable. The second equation is easiest: add y to both sides and subtract 1 to get y = x - 1.

Step 2 — Substitute. Replace y in the first equation with x - 1: 2x + (x - 1) = 5.

Step 3 — Solve the single-variable equation. Combine like terms: 3x - 1 = 5, so 3x = 6, giving x = 2.

Step 4 — Back-substitute. Put x = 2 into y = x - 1: y = 2 - 1 = 1.

Step 5 — Check. Test (2, 1) in both original equations: 2(2) + 1 = 5 ✓ and 2 - 1 = 1 ✓. The solution is (2, 1).

The substitution method in five steps

1. Pick the easiest variable to isolate. Look for a coefficient of 1 or -1. Example: in x - y = 1, solving for y gives y = x - 1 with no fractions.
2. Substitute that expression into the other equation. Never plug back into the same equation you just used — that gives 0 = 0 and no information.
3. Solve the resulting one-variable equation. Distribute, combine like terms, and isolate the variable. Example: 2x + (x - 1) = 5 becomes x = 2.
4. Back-substitute to find the other variable. Use the isolated expression from step 1. Example: y = 2 - 1 = 1.
5. Write the solution as an ordered pair and check it. Both original equations should be true.

Another example with different numbers

Solve the system:

• y = 3x - 4
• 2x + y = 6

Step 1 — Isolate a variable. The first equation already has y isolated: y = 3x - 4.

Step 2 — Substitute. Replace y in the second equation: 2x + (3x - 4) = 6.

Step 3 — Solve. Combine: 5x - 4 = 6, so 5x = 10, giving x = 2.

Step 4 — Back-substitute. y = 3(2) - 4 = 6 - 4 = 2.

Step 5 — Check. y = 3(2) - 4 = 2 ✓ and 2(2) + 2 = 6 ✓. The solution is (2, 2).

Common mistakes to avoid

• Substituting into the same equation you solved. If you solved equation 1 for y, you must plug into equation 2 — otherwise you'll get 0 = 0, which is true but useless.
• Forgetting parentheses around the substituted expression. Writing 2x + x - 1 instead of 2x + (x - 1) is fine here, but when the expression has a coefficient in front — like -3(x - 1) — dropping the parentheses gives the wrong sign on the constant.
• Stopping after finding one variable. A system's solution is an ordered pair. If you only report x = 2, you're only halfway done.
• Not simplifying before substituting. Isolating a variable when its coefficient is 3 or 5 first forces fractions into every later step. Pick the equation and variable with coefficient ±1 whenever possible.
• Skipping the check. A quick substitution back into both originals catches sign errors instantly.

Practice this method with a guided tutor

The steps above show you the mechanics, but substitution really sticks once you've worked several problems where the tutor catches your slip the moment it happens. Sign up for LernOS to try our guided Socratic tutor — it walks you through systems like these one nudge at a time, so you build the instinct for which variable to isolate first.