How to solve a system of two equations

Every step

1. 1

   Choose a variable to isolate

   Write x + y = 5 and 2x − y = 1 on separate lines. In the first equation, subtract x from both sides to get y = 5 − x. Substitution works well here because the coefficient of y is 1. You will replace y in the second equation with this whole expression.

   CHECKPOINT · You have y = 5 − x and the unchanged second equation.
2. 2

   Substitute and solve for one variable

   Plug y = 5 - x into the second equation: 2x - (5 - x) = 1. This simplifies to 2x - 5 + x = 1. Combine like terms: 3x - 5 = 1; then add 5: 3x = 6; divide by 3: x = 2.

   CHECKPOINT · Value for one variable found (e.g., x = 2).
3. 3

   Solve for the other variable

   Substitute x = 2 into y = 5 - x: y = 5 - 2 = 3. Thus, the solution is x = 2, y = 3.

   CHECKPOINT · Both variables have values.
4. 4

   Check both equations with your solution

   For x + y = 5: 2 + 3 = 5 (correct). For 2x - y = 1: 2*2 - 3 = 4 - 3 = 1 (correct). Both check out, so your solution is valid.

   CHECKPOINT · Substituted values satisfy both equations.
5. 5

   Use elimination when terms cancel

   For 2x + 3y = 12 and x − 3y = 3, add the equations: 3x = 15, so x = 5. Substitute into x − 3y = 3: 5 − 3y = 3, so y = 2/3. Check both original equations: 10 + 2 = 12 and 5 − 2 = 3.

   CHECKPOINT · Both values satisfy both equations.