Equation of a Line (y = mx + c)

Statement

The equation of a straight line in slope–intercept form is:

\[ y = mx + c \]

Here, \(m\) is the slope (gradient) of the line, and \(c\) is the y-intercept (the value of \(y\) when \(x=0\)).

Why it’s true

• The slope \(m\) tells us how much \(y\) changes for a unit change in \(x\): \(\Delta y/\Delta x = m\).
• The intercept \(c\) fixes where the line crosses the y-axis.
• Together, slope and intercept uniquely define any non-vertical line in the plane.

Recipe (how to use it)

1. Identify the slope \(m\).
2. Find the y-intercept \(c\) (where the line meets the y-axis).
3. Write equation as \(y=mx+c\).

Spotting it

This form is common in graphs of linear equations. If you see a line, its slope and y-intercept can be read directly to form the equation.

Common pairings

• Gradient formula: \(m=\frac{y_2-y_1}{x_2-x_1}\).
• Point–slope form \(y-y_1=m(x-x_1)\) (convertible into slope–intercept form).
• General line form \(ax+by+c=0\).

Mini examples

1. Slope: 2, Intercept: -3 → \(y=2x-3\).
2. Slope: -½, Intercept: 4 → \(y=-\tfrac{1}{2}x+4\).
3. Slope: 0, Intercept: 7 → \(y=7\) (horizontal line).

Pitfalls

• Mixing up slope with intercept.
• For vertical lines, slope is undefined—this form cannot be used.

Exam strategy

• If given two points, calculate slope first and then substitute to find intercept.
• Sketch by plotting the intercept and using the slope to find another point.

Summary

The slope–intercept equation \(y=mx+c\) is the most direct way to describe a line: \(m\) controls its steepness, and \(c\) tells where it crosses the y-axis.