Slope and Linear Equations: y = mx + b Explained

A linear equation graphs as a straight line. The most useful form is y = mx + b, where m is the slope and b is the y-intercept.

Slope (m)

Slope measures how steep a line is. It's "rise over run" — how much y changes when x increases by 1.

Example: Find slope between (1, 2) and (4, 8)

Slope = 2 means: for every 1 unit right, the line goes up 2 units.

Types of Slope

• Positive — line rises left to right
• Negative — line falls left to right
• Zero — horizontal line (y = constant)
• Undefined — vertical line (x = constant)

y-Intercept (b)

The y-intercept is where the line crosses the y-axis (when x = 0). In y = mx + b, b is that value.

Example: y = 3x + 5 → slope = 3, y-intercept = 5. The line crosses the y-axis at (0, 5).

Writing the Equation from Two Points

Example: Line passes through (2, 3) and (4, 7)

Step 1: Find slope.

Step 2: Use one point and slope in y = mx + b to find b.

Equation: y = 2x − 1

Point-Slope Form

When you know one point (x₁, y₁) and the slope m:

Example: Line through (3, 4) with slope 5 → y − 4 = 5(x − 3). You can simplify to y = 5x − 11.

Parallel and Perpendicular Lines

• Parallel lines have the same slope (m₁ = m₂)
• Perpendicular lines have slopes that multiply to −1: m₁ × m₂ = −1

Example: A line with slope 2 is perpendicular to a line with slope −½, since 2 × (−½) = −1.

Common mistakes

Rise and run order: Slope = rise / run = (y&sub2; − y&sub1;) / (x&sub2; − x&sub1;). Don't swap x and y.

Negative slope: If the line goes down from left to right, the slope is negative. Make sure you preserve the negative sign.

More examples

Example: Find the slope between (2, 3) and (6, 11).

Example: Write the equation of a line with slope 3 passing through (1, 5).

Try Arithmia