Linear Equations: Slope, Y-Intercept, and Rate of Change
How to graph straight lines, interpret slope as a practical rate of change, and find parallel or perpendicular lines.
The Slope-Intercept Form
The most widely used format for linear equations is the slope-intercept form:
Where m represents the slope (steepness and direction) and b represents the y-intercept (the point (0, b) where the line crosses the vertical axis).
Calculating Slope (Rise Over Run)
Slope measures the vertical change (Δy) divided by the horizontal change (Δx) between any two points (x&sub1;, y&sub1;) and (x&sub2;, y&sub2;) on the line:
A taxi charges a $3.50 base pickup fee plus $2.25 per mile driven.
• Slope (m) = $2.25/mile (rate of change)
• Y-intercept (b) = $3.50 (starting cost at 0 miles)
• Equation: y = 2.25x + 3.50
For an 8-mile trip: y = (2.25 × 8) + 3.50 = $18.00 + $3.50 = $21.50.
Standard Form and General Form (Ax + By = C and Ax + By + C = 0)
While slope-intercept form ($y = mx + b$) is convenient for graphing, linear equations are commonly encountered in Standard Form ($Ax + By = C$) or General Form ($Ax + By + C = 0$).
You can determine the slope directly from the coefficients without manual algebra using the standard ratio:
For example, for 3x − 2y = 8, the slope is m = −(3) ÷ (−2) = 1.5, with y-intercept (0, C/B) = (0, −4).
Parallel vs. Perpendicular Lines
• Parallel lines have identical slopes: m&sub1; = m&sub2;. They maintain a constant distance and never meet.
• Perpendicular lines intersect at right angles (90°). Their slopes are negative reciprocals: m&sub1; × m&sub2; = −1 (or m&sub2; = −1/m&sub1;). If line A has slope 3, line B has slope −1/3.