AI Tutor
If two highway routes intersect in the middle of nowhere, do you actually have to calculate their exact GPS intersection coordinates just to design a third road that passes through that same junction?
In Cartesian geometry, the answer is no. By understanding how the coefficients of a linear equation control its orientation, you can manipulate entire families of lines at once.
Before we look at entire bundles of lines, how do we test whether two individual lines never meet or cross at right angles using just their standard equations?

Gradients and the General Form

Any non-vertical straight line can be written in general form as Ax+By+C=0Ax + By + C = 0, where AA, BB, and CC are real constants and AA and BB are not both zero.
Rearranging this equation into slope-intercept form gives y=βˆ’ABxβˆ’CBy = -\frac{A}{B}x - \frac{C}{B}, which reveals that the gradient (the slope mm) is always m=βˆ’ABm = -\frac{A}{B}.
Two lines L1:A1x+B1y+C1=0L_1: A_1 x + B_1 y + C_1 = 0 and L2:A2x+B2y+C2=0L_2: A_2 x + B_2 y + C_2 = 0 are parallel (L1βˆ₯L2L_1 \parallel L_2) if and only if their slopes are equal, meaning βˆ’A1B1=βˆ’A2B2-\frac{A_1}{B_1} = -\frac{A_2}{B_2}.
Cross-multiplying gives the general determinant criterion for parallelism: A1B2βˆ’A2B1=0A_1 B_2 - A_2 B_1 = 0, meaning the coefficients of xx and yy are directly proportional.

Why Perpendicular Slopes Multiply to -1

When René Descartes introduced Cartesian coordinates in 1637, he linked Euclidean geometry to algebra by showing that a 90∘90^\circ rotation swaps horizontal and vertical displacements while reversing one sign.
If a line has direction vector (1,m1)(1, m_1), rotating it by 90∘90^\circ counterclockwise yields the vector (βˆ’m1,1)(-m_1, 1), whose slope is m2=1βˆ’m1=βˆ’1m1m_2 = \frac{1}{-m_1} = -\frac{1}{m_1}.