When Do Two Linear Equations Have Infinitely Many Solutions?

Quick answer: Infinitely many solutions occur when two equations describe the same line. Multiplying all coefficients and the constant by the same nonzero factor leaves the equation unchanged.

Worked example

Compare 2x + 3y = 12 and 4x + 6y = 24.

1. Divide the second equation by 2.
2. It becomes exactly 2x + 3y = 12.
3. Let y = t; then x = (12 − 3t)/2, for any real t.

Check the result

For t = 0, (6, 0) works. For t = 2, (3, 2) works. There is a whole line of solutions, not just these two.

Common mistake to avoid

A subtraction result 0 = 0 is an identity, not x = 0 or y = 0. If division by a zero coefficient would occur, compare equations by elimination.

Try it yourself

1. Test x − y = 3 and 3x − 3y = 9. 2. Does x − y = 3 together with 3x − 3y = 10 have infinitely many solutions?

Answers and reasoning

1. Yes: (x, y) = (t + 3, t). 2. No: tripling the first requires 9 on the right, so the pair is inconsistent.

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