Slope-intercept form: y = mx + b

In y = −3x + 5 you can read the graph straight off the equation: start at (0, 5), then go 1 right and 3 down.

Two numbers that describe a line

When a linear equation is solved for y, it takes the shape y = mx + b. OpenStax calls this the slope-intercept form, because the two numbers in it are exactly the two facts you need to draw the line. The coefficient of x, m, is the slope. The constant, b, is the y-coordinate of the y-intercept, so the line crosses the y-axis at (0, b). For y = −3x + 5 the slope is −3 and the y-intercept is (0, 5). No table is needed: the equation already tells you where to start and which way to go.

To graph from the form, plot the y-intercept first. Then write the slope as a fraction rise/run and count it out from that point. A slope of −3 is −3/1: one unit right and three units down, which lands on (1, 2). Do it again to reach (2, −1), and draw the line through the points. For y = (2/5)x − 1, start at (0, −1), go 5 right and 2 up to (5, 1). A negative slope can be counted either as down-then-right or as up-then-left; both land on the same line.

If an equation is not in this form, solve it for y first. OpenStax works through x + 2y = 6: subtract x from both sides to get 2y = −x + 6, then divide by 2 to get y = −(1/2)x + 3. Now you can read the slope, −1/2, and the y-intercept, (0, 3). This is the same line you graphed with intercepts in the previous unit, and the x-intercept (6, 0) agrees: start at (0, 3) and go 2 right and 1 down three times.

What m and b mean in a formula

The form works with any letters. The textbook formula F = (9/5)C + 32 converts Celsius to Fahrenheit. Its slope, 9/5, says Fahrenheit goes up 9 degrees for every 5 degrees Celsius. Its intercept, 32, says that 0 °C is 32 °F. Another OpenStax example models a delivery van: C = 0.5m + 60, where C is the weekly cost in dollars and m is miles driven. The 60 dollars is the fixed cost you pay even with no driving, and the 0.5 is the cost of each extra mile. A week of 250 miles costs 0.5 × 250 + 60 = 185 dollars.

Parallel and perpendicular lines

Two lines with the same slope and different y-intercepts never meet: they are parallel. So y = 2x + 1 and y = 2x − 5 are parallel, and you can tell without drawing them. Perpendicular lines meet at a right angle, and their slopes are negative reciprocals: flip the fraction and change the sign. OpenStax shows slopes 1/4 and −4, whose product is −1. That product is always −1 for two perpendicular lines, as long as neither is vertical. A horizontal line and a vertical line are also perpendicular, but the vertical one has no slope to multiply.