A system of equations is a set of equations that must all be true at the same time. For two linear equations in two variables, the solution is the point where the two lines cross.
Elimination
Add or subtract the equations so that one variable cancels.
x + y = 10 and x - y = 4
x + y = 10 + x - y = 4 ------------ 2x = 14 x = 7 substitute back: 7 + y = 10, so y = 3 solution: x = 7, y = 3
Check both original equations: 7 + 3 = 10 and 7 - 3 = 4. Checking in only one equation is not a check.
Substitution
When one equation already gives a variable on its own, substitution is faster.
y = 2x + 1 and 3x + y = 11
3x + (2x + 1) = 11 5x + 1 = 11 5x = 10 x = 2 y = 2(2) + 1 = 5 solution: x = 2, y = 5
Elimination when nothing cancels yet
Multiply one or both equations first so that a variable will cancel.
2x + 3y = 16 and x - y = 3
multiply the second by 2: 2x + 3y = 16 2x - 2y = 6 ------------ subtract 5y = 10 y = 2 x - 2 = 3, so x = 5 solution: x = 5, y = 2
Graphing
Draw both lines and read off the crossing point. Graphing is the method that shows what a solution is, and it is the least accurate for anything that does not land on a grid intersection. Use it to understand, then use algebra to answer.
No solution, or infinitely many
| Result when solving | Meaning | The lines |
|---|---|---|
| one value for each variable | exactly one solution | cross once |
| a false statement such as 0 = 6 | no solution | are parallel and never meet |
| a true statement such as 0 = 0 | infinitely many solutions | are the same line |
Common mistakes
Sign errors when subtracting equations
Subtracting an equation means subtracting every term of it, including the right hand side.
Stopping after finding one variable
A solution is a pair. Finding x and not substituting back for y is an unfinished answer.
Check yourself
Solve x + y = 12, x - y = 2.
Adding gives 2x = 14, so x = 7 and y = 5.
Solve y = 3x, x + y = 16.
Substituting, x + 3x = 16, so x = 4 and y = 12.
Solve 3x + 2y = 18, x = 4.
12 + 2y = 18, so y = 3.
What does it mean if solving gives 0 = 5?
The system has no solution. The lines are parallel, so there is no point on both of them.
See also
Sources
- Systems of linear equations as taught in United States eighth grade and introductory algebra. Examples on this page are our own.