Where two lines meet
You know one line: y = mx + b. A system of equations is just two lines at once. The solution is the point that sits on both.
1. Move the lines, watch the solution
Drag the sliders. Each line has its own slope (m) and y-intercept (b). The gold dot is the solution, if there is one.
2. Solve it with algebra
At the solution, both lines have the same y. So set the two right-hand sides equal and solve for x. This is the substitution method.
Try the “Parallel lines” and “Same line” buttons above and watch what happens to the steps.
3. Three possible outcomes
The tab follows the lines you set in part 1. You can also click any tab to read about that case.
The lines cross exactly once. Their slopes are different, so they can’t run side by side forever.
Example: y = 2x − 1 and y = −x + 5 meet at (2, 3).
The lines never meet. They have the same slope but different y-intercepts, so they are parallel. No point sits on both, so the system has no solution.
Example: y = x + 2 and y = x − 3. Both climb 1 for every 1 across, but one always sits 5 higher.
The two equations describe the same line. Same slope and same y-intercept, so every point on one line is also on the other. That is infinitely many solutions.
The equations can look different. y = 2x + 1 and 2y = 4x + 2 are the same line, because dividing the second by 2 gives the first.
A quick test: compare m₁ and m₂ first. Different slopes always means one solution. Same slopes means you check the intercepts.
4. Practice
Pick a kind of problem. Your first answer on each problem counts toward your progress.
Find the point where these lines cross. Enter x and y, then check.
Without solving, decide how many solutions this system has.
5. Progress
Only your first answer on each problem counts. A problem type shows “mastered” after 5 problems at 80% or better. Progress is saved in this browser only.
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Quiz and test
Pick a level and a mode. Practice gives instant feedback and never ends. A test is 10 questions, with your score and a review of misses at the end.