Start for Free

Algebra guide

How to solve systems of equations

Substitution, elimination, and graphing for two-variable systems, with worked examples, answer checks, and how to tell when there is no solution.

A system of equations is two or more equations that share the same variables. The solution is the set of values that makes every equation true at once. For two linear equations in x and y, that is the point where the two lines cross.

Method 1: Substitution

Best when one equation already isolates a variable. Solve y = 2x + 1 and 3x + y = 11.

  1. Replace y in the second equation: 3x + (2x + 1) = 11
  2. Simplify: 5x + 1 = 11, so 5x = 10 and x = 2.
  3. Back-substitute: y = 2(2) + 1, so y = 5.

Check: 3(2) + 5 = 11 ✓.

Method 2: Elimination

Best when coefficients line up. Solve x + y = 10 and x − y = 4. Add the equations so y cancels: 2x = 14, so x = 7 and y = 3. Check: 7 + 3 = 10 ✓ and 7 − 3 = 4 ✓.

When nothing cancels yet, multiply first. Solve 2x + 3y = 12 and 4x − y = 10.

  1. Multiply the second equation by 3: 12x − 3y = 30
  2. Add it to the first: 14x = 42, so x = 3.
  3. Substitute: 2(3) + 3y = 12, so 3y = 6 and y = 2.

Check: 4(3) − 2 = 10 ✓.

How many solutions?

  • One: the lines have different slopes and cross once.
  • None: same slope, different intercepts (parallel lines). Elimination ends in a false statement such as 0 = 5.
  • Infinitely many: the equations describe the same line. Elimination ends in 0 = 0.

Common mistakes

  • Multiplying only one side, or only some terms, of an equation.
  • Stopping after finding x and never solving for y.
  • Skipping the check in both original equations.

More study guides