Equation of a Line
309. Slope-Intercept Form · y = mx + c revealed
The ratio of vertical rise to horizontal run always equals the magnitude of the slope m.
The slope–intercept form of a line is y = mx + c, where m is the slope (rise over run) and c is the y-intercept. Reading m and c straight off the equation gives the line's steepness and where it crosses the y-axis.
What this lesson covers
Try to break it
Drag P along the line. The right triangle below P has vertical leg (P_y − c) and horizontal leg (P_x − 0). Their ratio is always the slope m, so y − c = mx — rearranged, y = mx + c. Every point on the line satisfies the equation; try to find one that doesn't.
How you build it
Build the line y = mx + c from its intercept and slope.
- On the y-axis, count 3 units up from the origin O and mark point B at (0, 3). This is the intercept c = 3.
- From B, step the slope m = ¾: count 4 units right (the run) and 3 units up (the rise) and mark P at (4, 6).
- Draw the straight line through B and P. It crosses the y-axis at c = 3 and rises ¾ for every 1 across — the graph of y = ¾x + 3.
The proof, step by step
Prove that the line y = mx + c has slope m and y-intercept c.
- Identify the y-intercept B at (0, c) and any point P at (x, y) on the line.
- Form right triangle BPR by dropping a perpendicular from P to the y-axis at R.
- The vertical leg BR has length y - c, and the horizontal leg RP has length x.
- Slope m = tan θ = (vertical rise) / (horizontal run) = (y - c) / x.
- Rearrange the equation: mx = y - c, which gives y = mx + c.
Worked example
The equation of a line with slope 2 and y-intercept 5 is:
Using the slope-intercept form y = mx + c, substitute m = 2 and c = 5 to directly obtain y = 2x + 5.
- y = 2x + 5 — correct
- y = 5x + 2
- y = 2x - 5
- y = -2x + 5