SAT Math · Algebra

Slope-intercept form

Q: What is slope-intercept form?

Slope-intercept form is y = mx + b, a way of writing a linear equation so you can read its graph directly. m is the slope, and b is the y-intercept, the point (0, b) where the line crosses the y-axis.

Q: How do you find the slope from two points?

Divide the change in y by the change in x: m = (y₂ − y₁) ÷ (x₂ − x₁). For the points (2, 3) and (5, 9), the slope is 6 ÷ 3 = 2.

Q: How do you convert standard form to slope-intercept form?

Solve the equation for y. For 3x + 2y = 6, subtract 3x from both sides to get 2y = −3x + 6, then divide by 2 to get y = −3/2 x + 3.

Q: What is the slope of a horizontal or vertical line?

A horizontal line like y = 5 has slope 0, because y never changes. A vertical line like x = 2 has an undefined slope, because the run is zero and you cannot divide by zero.

Q: What does the slope mean in a word problem?

The slope is a rate: how much the output changes for each one-unit increase in the input. In C = 15h + 40, the cost goes up $15 for every additional hour.

Q: How can Aristotle help with SAT algebra?

Aristotle is a voice AI SAT tutor. Your child talks through a linear equation out loud, and Aristotle asks guiding questions instead of giving the answer, starting from the step where their reasoning broke. It tracks 113 SAT Math skills, including linear equations and functions.

Read a line's slope and y-intercept straight from y = mx + b, and write one from a graph or two points.

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Updated October 30, 2026

$y = mx + b$

$y = 2x + 1$ crosses the $y$ -axis at 1, then rises 2 for every 1 to the right.

What is slope-intercept form?

y = \underbrace{m}_{\text{tilt}}\,x + \underbrace{b}_{\text{start}}

A line's equation written like directions. Start at height $b$ on the $y$ -axis. Then for every 1 step to the right, go up $m$ . The slope sets the tilt; the intercept sets where the line begins.

Cross the y-axis at 1, then rise 2 for every 1 to the right.

Slope is the same between any two points

m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{8 - 2}{4 - 1} = 2

Pick any two points. The rise is the difference in their $y$ -values, the run is the difference in their $x$ -values, and slope is rise over run. Every pair gives the same number; that constant ratio is what makes a line straight.

From (1, 2) to (4, 8): run 3, rise 6, slope 2.

In a word problem, b is the start and m is the rate

C = \underbrace{2.5}_{\text{per mile}}\,d + \underbrace{3}_{\text{to get in}}

$b$ is what you pay before anything happens; $m$ is what each unit adds. A taxi that charges $3 to get in and $2.50 per mile costs $C = 2.5d + 3$ for $d$ miles. The 3 is the intercept and the 2.5 is the slope.

Starts at $3, climbs $2.50 for each mile.

What m and b mean

Answer each question in your head, then scroll for the answer.

Think it through

Question 1
In $y = 2x + 1$ , what is $y$ when $x = 0$ ?
Answer
$y = 1$ . That is the $y$ -intercept, $b$ .
Question 2
Now add 1 to $x$ . How much does $y$ change?
Answer
It goes up by 2, the number in front of $x$ . That is the slope, $m$ .
Question 3
In $C = 15h + 40$ , what do 15 and 40 mean?
Answer
$15 more per hour, plus a $40 fee before any hours.

Try it yourself

Work it out first, then check your answer. SAT-style questions, not official College Board questions.

Question 1

What are the slope and $y$ -intercept of the line $3x + 2y = 6$ ?

Question 2

A line passes through $(2, 3)$ and $(5, 9)$ . What is its equation in slope-intercept form?

Question 3

A plumber charges $C = 15h + 40$ dollars for a job that takes $h$ hours. What does the 40 represent?

Question 4

Line $k$ is perpendicular to $y = 2x + 1$ and passes through $(0, 4)$ . What is the equation of line $k$ ?

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The formulas

Slope-intercept form

$y = mx + b$

$m$ is the slope, $b$ is the $y$ -intercept. Solve for $y$ to get here.

Slope from two points

$m = \dfrac{y_2 - y_1}{x_2 - x_1}$

Rise over run. Subtract in the same order on the top and the bottom.

Point-slope form

$y - y_1 = m(x - x_1)$

Useful when you know the slope and one point. Distribute and solve for $y$ to reach slope-intercept form.

Parallel and perpendicular

$m_2 = m_1 \quad\text{or}\quad m_1 \cdot m_2 = -1$

Parallel lines share a slope. Perpendicular slopes are negative reciprocals, like 2 and $-\tfrac{1}{2}$ .

What the slope tells you
Slope $m$	The line	Example
$m > 0$	rises left to right	$y = 3x - 2$
$m < 0$	falls left to right	$y = -x + 4$
$m = 0$	horizontal	$y = 5$
undefined	vertical, not a function	$x = 2$

Try it

Tilt and slide the line

Tilt the line with m and slide it with b.

Slope m2

y-intercept b1

Equationy = 2x + 1

Slope2: up 2 for every 1 to the right

y-intercept(0, 1)

x-intercept(−0.5, 0)

Common mistakes

Not solving for y first

In $3x + 2y = 6$ the slope is not 3. Get $y$ alone first.

Flipping rise and run

Slope is change in $y$ over change in $x$ . Flipping it gives a common wrong answer.

Mixing up the subtraction order

Subtract in the same order on top and bottom, or the sign flips.

Using the opposite, not the negative reciprocal

Perpendicular to slope 2 is $-\tfrac{1}{2}$ , not $-2$ . Flip the fraction and change the sign.

How it shows up on the SAT

Algebra is about 35% of SAT Math. Expect slope from points or a table, matching a graph to its equation, and saying what m or b means in context.

What Aristotle tracks in linear equations in two variables

9 SAT Math skills, including:

Writing and using linear equations with two variables
Writing two-variable linear equations for constraints
Explaining parts and solutions of two-variable equations in context
Interpreting graphs of equations in Ax+By=C form

What is slope-intercept form?

Slope-intercept form is y = mx + b, a way of writing a linear equation so you can read its graph directly. m is the slope, and b is the y-intercept, the point (0, b) where the line crosses the y-axis.

How do you find the slope from two points?

How do you convert standard form to slope-intercept form?

What is the slope of a horizontal or vertical line?

What does the slope mean in a word problem?

How can Aristotle help with SAT algebra?

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