What you are looking for
A system of two linear equations asks for the one pair (x, y) that makes both equations true at the same time. Graphically, that is the point where the two lines cross.
Substitution — best when one variable is already alone
Solve: y = 2x + 1 and 3x + y = 11.
The first equation already gives y, so substitute it into the second: 3x + (2x + 1) = 11. Combine: 5x + 1 = 11, so 5x = 10 and x = 2. Put x back into the first equation: y = 2(2) + 1 = 5. The solution is (2, 5).
Always check by substituting into the equation you did not use: 3(2) + 5 = 11. Correct.
Elimination — best when the same variable lines up
Solve: 2x + 3y = 12 and 2x - y = 4.
The x terms match, so subtract the second equation from the first: (3y - (-y)) = 4y and 12 - 4 = 8, giving 4y = 8, so y = 2. Substitute back: 2x - 2 = 4, so x = 3. The solution is (3, 2).
When nothing lines up, multiply one or both equations first. For 3x + 2y = 16 and 5x - 4y = 2, double the first equation to get 6x + 4y = 32, then add: 11x = 34.
How to choose in five seconds
Use this rule and stop debating with yourself.
- A variable already isolated (y = … or x = …) → substitution.
- A variable with matching or easily matched coefficients → elimination.
- Coefficients of 1 anywhere → substitution is usually quicker.
- Asked for an approximate answer or a picture → graph both lines.
No solution and infinite solutions
If every variable cancels and you are left with something false, like 0 = 7, the lines are parallel and there is no solution. If you are left with something always true, like 0 = 0, the two equations describe the same line and there are infinitely many solutions.