How Many Solutions?
Drag the four points and watch two lines answer the same question three different ways: cross once, never meet, or turn out to be the same line drawn twice.
1 solution: (1, 1)
Different slopes: the lines cross exactly once, at the marked point.
- Line 1 (blue)
- y = x
- Line 2 (warm)
- y = −x + 2
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Ready to try some? Systems of equations worksheets
Where it usually goes wrong
Substitution and elimination are practiced as procedures that always end in a single (x, y) pair, so a system that comes out "0 = 0" or "3 = 7" reads as a mistake in the algebra rather than as a true answer about the lines themselves — nobody has seen what those two outcomes actually look like drawn, so there is nothing to recognize them against.
Try this
- Press "Cross once". Drag either line's points and watch the marked point slide along the OTHER line, never off it — that dot is the one pair of numbers both equations agree on.
- Press "Never meet". Try to drag one line until it touches the other. It cannot happen without changing the slope — same steepness, different heights, is exactly what "parallel" and "no solution" mean together.
- Press "Same line twice". Drag one of its points anywhere that keeps the two equations printed identical. Every point on the line now solves both equations, which is why the count reads infinite.
Where this goes next
Elimination that produces "0 = 0" is this picture's coincident case — the second equation was never new information — and "3 = 7" is the parallel case: the algebra correctly proves no such point exists rather than failing to find one.