System of linear equations — parallel lines

A system of two equations is two lines

Every linear equation $y = ax + b$ is a line in the plane. A system of two such equations describes two lines:

A solution of the system is a pair $(x, y)$ that satisfies both equations at once — that is, a common point of the two lines. So the number of solutions equals the number of points the lines share.

Three possible cases:

The lines intersect ($a_1 \neq a_2$) — they have exactly one common point → one solution. The intersection point:

Parallel, distinct lines ($a_1 = a_2 \;\wedge\; b_1 \neq b_2$) — they have the same slope but are shifted vertically, so they never meet. The solution set is empty — the system is inconsistent, no solutions.

The lines coincide ($a_1 = a_2 \;\wedge\; b_1 = b_2$) — it is the same line, every point of which is shared → infinitely many solutions (an undetermined system).

How do you solve such a system?

Substitution method — isolate one variable from one equation and plug it into the other. Elimination method — multiply the equations so that adding them cancels one variable.

For a system in general form you can also use a determinant. Writing the lines as $a_1x - y = -b_1$ and $a_2x - y = -b_2$, the main determinant is:

$W \neq 0$ (that is, $a_1 \neq a_2$) → exactly one solution. When $W = 0$, the system is either inconsistent (none) or undetermined (infinitely many) — comparing the intercepts settles it.

Why this matters for the exam:

Before you start computing, compare the slopes. Equal slopes are a signal: either no solutions or infinitely many — the intercept then decides the outcome. This saves time and guards against mistakes.

Remember: the number of solutions of a system = the number of points the lines share. Parallel and distinct ($a_1=a_2,\; b_1\neq b_2$) → the solution set is empty, the system is inconsistent.