Parallel and identical lines

Two straight lines can meet in exactly one point, never meet, or lie on top of each other. So a pair of simultaneous linear equations has exactly one solution, no solution, or infinitely many solutions.

Why: lines with different gradients must cross exactly once. Lines with the same gradient rise or fall at the same rate, so they keep the same vertical gap: if the gap is not zero they never meet, and if it is zero they are the same line.

Method

1. Rewrite each equation in the form \(y=mx+c\).
2. Compare the gradients. If they are different, there is exactly one solution.
3. If the gradients are equal, compare the \(y\)-intercepts: different intercepts give no solution; equal intercepts give infinitely many.
4. A quick test: if one whole equation, including the constant, is a multiple of the other, the lines are identical. If only the \(x\) and \(y\) terms are in that ratio, the lines are parallel.
5. Vertical lines cannot be written as \(y=mx+c\). Two vertical lines \(x=a\) and \(x=b\) with \(a\ne b\) are parallel, so there is no solution; a vertical line meets any line that is not vertical exactly once.

Example

Consider \(5x+y=4\) and \(10x+2y=3\). Rearranging gives \(y=-5x+4\) and \(y=-5x+1.5\). The gradients are both \(-5\) and the \(y\)-intercepts differ, so the lines are parallel and the equations have no solution.

Now consider \(5x+y=4\) and \(15x+3y=12\). The second equation is the first multiplied by \(3\), so both are the line \(y=-5x+4\). Every point on it is a solution, for example \((0,4)\) and \((1,-1)\).

Check: equal gradients alone do not decide the answer; you must also compare the intercepts or the constants. Two lines that only look parallel on a sketch may still meet far away, so compare the equations, not the drawing.