Functions
AI Assistant

Systems of Equations

When two lines meet, something interesting happens

One equation gives you a line — infinite points that all work. But what if you need a point that works for two equations at the same time? Now you're looking for where two lines cross.

In this lesson, you'll solve three different systems by finding crossing points on the graph. Each time, you'll check your answer by plugging it back in. Then comes the twist: what happens when two lines have the same slope? They never meet — and the system has no solution.

By the end, you'll understand the three possible outcomes: one solution (lines cross), no solution (parallel), or infinitely many (same line). All discovered visually, step by step.

Graph

FAQ

What is a system of equations?
A system of equations is a set of two or more equations with the same variables. For example, y = 2x + 1 and y = -x + 4 form a system. The solution is the (x, y) pair that makes BOTH equations true at the same time.
How many solutions can a system of two linear equations have?
A system of two linear equations can have one solution (the lines cross at one point), no solution (the lines are parallel — same slope, different intercept), or infinitely many solutions (the lines are identical — same slope AND same intercept).
How do parallel lines relate to systems with no solution?
Parallel lines have the same slope but different y-intercepts, so they never cross. For example, y = 2x + 1 and y = 2x - 1 are parallel. Since there is no intersection point, there is no (x, y) that satisfies both equations — the system has no solution.
How do I verify a solution to a system?
Plug the (x, y) values into BOTH equations and check that both are true. For example, if the solution is (1, 3): check equation 1: 2(1) + 1 = 3 ✓, check equation 2: -(1) + 4 = 3 ✓. Both work, so (1, 3) is correct.