Linear Equations: System of Two Lines

If the slopes are different ($a_1 \neq a_2$), the lines have different slopes and intersect at exactly one point. The system has one unique solution.

If the slopes are equal ($a_1 = a_2$), but the y-intercepts are different ($b_1 \neq b_2$), the lines run parallel to each other at a constant distance and never meet. The system has no solution.

If both the slopes and the y-intercepts are identical ($a_1 = a_2$ and $b_1 = b_2$), the two equations represent the exact same line. Every point on the line is a shared point, so the system has infinitely many solutions.

To enter subscript parameters like $a_1$ or $a_2$ (or $b_1$ and $b_2$) in the math input, type a_1 or a_2 using the underscore key (_) on your keyboard, or click the subscript button ($x_n$) on the virtual math keyboard. Note: Typing a1 or a2 without an underscore is not accepted, because in mathematics $a1$ represents $a \cdot 1$ and $a2$ represents $2a$.