AI Tutor
Ever wonder how a video game engine knows whether two laser paths will never meet, or if two walls meet at a crisp, perfect 90∘90^\circ corner?
It all comes down to slope, which measures a line's steepness as the vertical change over horizontal change: m=Ξ”yΞ”xm = \frac{\Delta y}{\Delta x}.
If two lines never meet no matter how far they stretch across the plane, what exact relationship must their slopes share?

Parallel Lines and Equal Slopes

Two distinct lines in a plane are parallel if and only if they have the exact same slope (m1=m2m_1 = m_2) and different yy-intercepts.
In 1637, French mathematician RenΓ© Descartes pioneered analytic geometry, proving that two lines with identical rates of change will maintain constant separation forever.
Let's find the equation of the line passing through (4,βˆ’1)(4, -1) that is parallel to 3xβˆ’2y=83x - 2y = 8.
First, isolate yy to find the slope: 2y=3xβˆ’8β€…β€ŠβŸΉβ€…β€Šy=32xβˆ’42y = 3x - 8 \implies y = \frac{3}{2}x - 4, so the given slope is m=32m = \frac{3}{2}.
Using point-slope form, yβˆ’y1=m(xβˆ’x1)y - y_1 = m(x - x_1), we substitute (x1,y1)=(4,βˆ’1)(x_1, y_1) = (4, -1) and m=32m = \frac{3}{2} to get yβˆ’(βˆ’1)=32(xβˆ’4)y - (-1) = \frac{3}{2}(x - 4), which simplifies to 3xβˆ’2y=143x - 2y = 14.
What happens if we take that same line and rotate it by exactly 90∘90^\circ instead?

Perpendicular Lines and Negative Reciprocals