AlgebraAlgebra 1High school

Slope and Lines

Slope, slope-intercept and point-slope form, parallel and perpendicular lines.

Slope measures how steep a line is: the change in $y$ divided by the change in $x$ between any two points, $m = \frac{y_2 - y_1}{x_2 - x_1}$ . Because a line has constant steepness, every pair of points gives the same slope, which is what makes a straight line straight.

Two forms of a line's equation cover almost every task. Slope-intercept form $y = mx + b$ shows the slope and the $y$ -intercept directly. Point-slope form $y - y_1 = m(x - x_1)$ lets you write the equation the moment you know one point and the slope.

Key terms

Slope: Rise over run between two points on a line.
$m = \frac{y_2 - y_1}{x_2 - x_1}$
Perpendicular slopes: Non-vertical perpendicular lines have slopes whose product is -1.
$m_1 m_2 = -1$

Key ideas

Positive slope rises left to right, negative slope falls, zero slope is horizontal, and a vertical line has undefined slope.
Parallel lines have equal slopes.
Perpendicular lines have slopes whose product is $-1$ , such as $2$ and $-\frac{1}{2}$ .
Interpret slope as a rate: units of $y$ per unit of $x$ .

Worked example

Find the equation of the line through $(2, 5)$ and $(6, -3)$ .

Slope: $m = \frac{-3 - 5}{6 - 2} = \frac{-8}{4} = -2$ .
Point-slope with $(2, 5)$ : $y - 5 = -2(x - 2)$ .
Expand: $y - 5 = -2x + 4$ .
Solve for $y$ : $y = -2x + 9$ .

Answer: $y = -2x + 9$

Common mistakes

Mixing the order of points: $\frac{y_2 - y_1}{x_1 - x_2}$ flips the sign.
Putting the change in $x$ on top.
Taking the negative instead of the negative reciprocal for a perpendicular slope.

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.Find the slope through $(1, 4)$ and $(5, 12)$ .
Show answer
$2$
2.What is the $y$ -intercept of the line with slope $3$ through $(2, 1)$ ?
Show answer
$-5$
3.What slope is perpendicular to $y = \frac{2}{3}x - 1$ ?
Show answer
$-\frac{3}{2}$

Slope and Lines

Key terms

Key ideas

Worked example

Common mistakes

Practice questions

Keep practising

Calculators

Worked problems

Video lessons

More in Algebra 1