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Elimination step by step, with both lines drawn and their crossing marked.
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The Coddy math course teaches the method itself - you work each step on an interactive board and get told exactly where a move went wrong.
Two equations in x and y are two lines on the same plane. A pair (x, y) that satisfies both equations is a point that lies on both lines - so it is the crossing. That is the whole idea, and it explains every case at once: lines that cross once give one solution, parallel lines give none, and a line drawn twice gives infinitely many.
Elimination and substitution are just ways of finding that crossing without drawing it. This page does both halves: the picture, so you can see what the answer means, and the elimination written out line by line, so you can see how it was found. Most calculators give you one or the other.
The two special cases are named rather than reported as failures. "No solution" and "infinitely many solutions" are real answers with real meanings, and a solver that shrugs at them teaches you to fear the question. Here the shape is worked out from the coefficients, so the page can tell you which case you are in even when there is no single pair to print.
Write them as you would on paper: 2x + y = 5 on the first line, x - y = 1 on the second. Fractions and decimals in the coefficients are fine.
The pair is at the top. Under it, a sentence says what the system is geometrically - one crossing, parallel, or the same line twice.
The lines are drawn with the crossing marked. If the answer surprises you, the picture usually explains it faster than the algebra does.
The numbered steps show which unknown was removed, how, and what was left. That is the method you are being asked to reproduce.
Every 2x2 system is one of these, and you can tell which from the lines before you do any algebra.
| Case | The lines | Example |
|---|---|---|
| One solution | Cross once | 2x + y = 5, x - y = 1 → (2, 1) |
| No solution | Parallel, different intercepts | x + y = 3, x + y = 5 |
| Infinitely many | The same line twice | x + y = 3, 2x + 2y = 6 |
| One equation only | A whole line of solutions | 2x + 3y = 7 |
| Three unknowns | Planes, not lines | x + y + z = 6, x - y = 0, z = 2 |
2x + y = 5; x - y = 1Adding the two equations removes y immediately, because +y and -y cancel: 3x = 6, so x = 2, and then y = 1 from either equation. The picture shows the two lines crossing at (2, 1) - one point, one answer.
x + y = 3; x + y = 5Subtract and you get 0 = -2, which no pair of numbers can fix. That is not the solver failing; it is the answer. The lines have the same slope and different intercepts, so they never meet and the system has no solution.
x + y = 3; 2x + 2y = 6The second equation is the first one doubled, so it draws the same line. Every point on that line satisfies both, which means infinitely many solutions. In the picture you only see one line, because there is only one.
x + y + z = 6; x - y = 0; z = 2Substituting z = 2 and x = y turns three equations into one, giving x = y = 2. There is no picture here - three unknowns are planes rather than lines - but the elimination is the same procedure, one unknown at a time.
