A system of equations is two (or more) equations that must be true at the same time. For two linear equations, each one is a straight line, and the solution is the point that lies on both lines: the place where they cross.
Step by step
- Write each equation in slope-intercept form, y = mx + b, so it is easy to draw. For example, x + y = 5 becomes y = −x + 5.
- Draw the first line. Plot its y-intercept, then use the slope to find a second point. For y = x + 1, start at (0, 1) and go up 1, right 1.
- Draw the second line the same way on the same axes. For y = −x + 5, start at (0, 5) and go down 1, right 1.
- Read the crossing point. Here the lines meet at (2, 3).
- Check it in both equations. y = x + 1: 3 = 2 + 1 ✓. y = −x + 5: 3 = −2 + 5 ✓.
See this system in the Intersection Finder →
Three possible outcomes
- The lines cross once: exactly one solution. This happens whenever the slopes are different.
- The lines are parallel: they never meet, so there is no solution. Same slope, different y-intercepts.
- The lines are the same: they overlap everywhere, so there are infinitely many solutions.
The last two cases have their own lesson: No solution or infinitely many solutions.
The weakness of graphing
Graphing is quick and shows what is going on, but you can only read the answer accurately when the lines cross at a grid point.
Example
Solve y = 2x − 1 and y = −x + 3 by graphing.
The lines cross a little to the right of x = 1 and a little below y = 2. From the graph you might estimate (1.3, 1.7).
The exact answer is (4/3, 5/3), which you can only get with algebra: 2x − 1 = −x + 3, so 3x = 4 and x = 4/3. Then y = 2 × 4/3 − 1 = 5/3.
See the exact answer in the Intersection Finder →
When the crossing point isn’t on a grid point, use substitution or elimination to find it exactly, and use the graph to check that your answer is sensible.