Two lines

Set the slope and intercept of the blue line and of the orange line. The dot marks the one point that lies on both, and the readout works out its coordinates by setting the two right-hand sides equal. Now press "make parallel": the dot vanishes and the algebra asks for something impossible. Press "make the same line" and every point works. Those are the only three things that can happen.

m₁ 1 b₁ -1

m₂ -0.5 b₂ 2

What's going on

A point (x, y) solves a system of two equations when it lies on both graphs. For two lines that means a crossing point. Setting m₁x + b₁ = m₂x + b₂ and collecting the x terms gives (m₁ − m₂)x = b₂ − b₁. If the slopes differ you can divide, and there is exactly one x, hence exactly one solution. Put that x back into either equation to get y.

If the slopes are equal you cannot divide by m₁ − m₂ = 0. Then the equation reads 0 = b₂ − b₁. When the intercepts differ this is false for every x: the lines are parallel and there is no solution. When the intercepts agree too, it is true for every x: the two equations describe the same line and there are infinitely many solutions.

So you can predict the outcome before solving anything: compare the slopes first, and then, only if they match, the intercepts. Equations that are not written as y = mx + b can hide this. Rearranging 2x + 4y = 8 into y = −½x + 2 shows what its slope is.

Sheets: Two equations, two unknowns and One, none, or infinitely many. Later: systems of curves.