Slope-Intercept Form

Slope-Intercept Form

The slope-intercept form of a line's equation is:

where m is the slope and b is the y-intercept — the value of y where the line crosses the y-axis (i.e., the point (0, b)).

Reading Slope and Intercept Directly

Because the equation is already solved for y, you can read off the slope and intercept immediately. In y = 3x - 4, the slope is 3 and the y-intercept is -4 (the line passes through (0,-4)).

Writing an Equation from a Graph

If you can see (or are told) two clear features of a line — where it crosses the y-axis, and how steep it is — you can write its equation directly. Identify b as the y-coordinate where the line crosses the y-axis, then find m by picking two points on the line and using the slope formula.

Writing an Equation from Two Points

Given two points, first compute the slope, then use one of the points to solve for b.

Example

Write the slope-intercept equation of the line through (2, 7) and (4, 13).

1. Find the slope: m = \frac{13 - 7}{4 - 2} = \frac{6}{2} = 3
2. Use the slope and one point, say (2, 7), in y = mx + b to solve for b: 7 = 3(2) + b \;\Rightarrow\; 7 = 6 + b \;\Rightarrow\; b = 1
3. Write the final equation using m=3 and b=1: y = 3x + 1

Check with the other point: y = 3(4)+1 = 13, which matches (4, 13).

Parallel and Perpendicular Hints

Slope-intercept form also makes it easy to compare lines at a glance. Two lines with the same slope m but different y-intercepts are parallel — they never cross, since they rise and fall at exactly the same rate. For instance, y = 2x + 1 and y = 2x - 5 are parallel. Recognizing this quickly, just by comparing the coefficient of x in two equations, is one of the most useful shortcuts that slope-intercept form provides.

Converting from Standard Form

An equation like 4x + 2y = 10 is in standard form, not slope-intercept form. To convert it, solve for y: subtract 4x from both sides to get 2y = -4x + 10, then divide everything by 2 to get y = -2x + 5. Now the slope (-2) and y-intercept (5) are immediately visible.

Key Takeaways

• Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.
• The line crosses the y-axis at (0, b).
• From two points, find m first, then substitute one point to solve for b.
• Always verify your equation works for both given points.