How do I find the slope and y-intercept if I only have two points?

Use the slope formula m = \frac{y_2 - y_1}{x_2 - x_1} to find the slope. Then substitute either point into y = mx + b and solve for b. If one of the points is on the y-axis (where x = 0), that point's y-coordinate is b directly.

What does it mean if the slope is 0?

A slope of 0 means the line is horizontal and does not rise or fall as x changes. The equation becomes y = b, a constant; every point on the line has the same y-value.

Can a line have a negative y-intercept?

Yes, absolutely. A negative b means the line crosses the y-axis below the origin, at the point (0, b) where b is negative. This is perfectly valid and very common in real-world situations.

If I have the equation 3x - y = 6, how do I rewrite it as slope-intercept form?

Solve for y: subtract 3x from both sides to get -y = -3x + 6, then divide every term by -1 to get y = 3x - 6. Now you can see directly that m = 3 and b = -6.

Slope-Intercept Form

Slope-intercept form directly reveals a line's slope and y-intercept, making it ideal for graphing and comparing linear relationships.

y = mx + b

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What each symbol means

What Slope-Intercept Form takes

Symbol	Meaning
$m$	The slope, which measures how steep the line is (rise over run); if $m$ is negative, the line slopes downward from left to right.
$b$	The y-intercept, the point where the line crosses the y-axis, occurring when $x = 0$ ; a negative $b$ means the line crosses below the origin.
$x$	The independent variable representing any point on the horizontal axis; each value of $x$ determines a corresponding $y$ value on the line.
$y$	The dependent variable representing the vertical coordinate; once you substitute $x$ , the equation tells you the exact $y$ value on the line.

When to use it

Reach for slope-intercept form when you need to quickly graph a line or write its equation from the slope and y-intercept.

Level

Usually taught in: Algebra I · Appears on: SAT, ACT

Worked examples

1. Convert slope and intercept to equation form

Problem

Write the equation of a line with slope 2 and y-intercept

-3

$m = 2, \quad b = -3$
Identify the slope $m$ and the y-intercept $b$ from the problem statement.
$y = mx + b$
Write the general slope-intercept form.
$y = 2x + (-3)$
Substitute $m = 2$ and $b = -3$ into the formula.
$y = 2x - 3$
Simplify by removing the parentheses.

Answer: $y = 2x - 3$

Slope-intercept form is the quickest way to turn slope and y-intercept into an equation because the variables directly replace $m$ and $b$ with no additional algebra needed.

2. Find equation from two points

Problem

A line passes through

(0, 5)

and

(4, 1)

. Write its equation in slope-intercept form.

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 5}{4 - 0}$
Use the slope formula with the two given points.
$m = \frac{-4}{4} = -1$
Simplify the fraction to find that $m = -1$ , meaning the line slopes downward.
$b = 5$
Since the line passes through $(0, 5)$ , the y-intercept is $b = 5$ .
$y = -x + 5$
Substitute $m = -1$ and $b = 5$ into slope-intercept form.

Answer: $y = -x + 5$

When you have two points, finding the slope first reveals the rate of change, and then identifying the y-intercept completes the equation in one clean substitution.

3. Model a player's season scoring with slope-intercept form

Problem

A basketball player scores at a constant rate. After 6 games, she has 84 total points. After 10 games, she has 140 total points. Write an equation relating total points scored (

y

) to the number of games played (

x

$(6, 84) \text{ and } (10, 140)$
Set up two ordered pairs from the given information, where each pair is (games, points).
$m = \frac{140 - 84}{10 - 6} = \frac{56}{4} = 14$
Calculate the slope using the two points; this means she scores 14 points per game on average.
$84 = 14(6) + b$
Substitute one point $(6, 84)$ and the slope $m = 14$ into $y = mx + b$ to solve for $b$ .
$84 = 84 + b \quad \Rightarrow \quad b = 0$
Simplify and solve to find that $b = 0$ , meaning she had 0 points at the start.
$y = 14x$
Substitute $m = 14$ and $b = 0$ into slope-intercept form.

Answer: $y = 14x$

This real-world scenario shows how slope-intercept form models linear relationships with a starting value ( $b$ ) and a constant rate of change ( $m$ ), making predictions and interpretations straightforward.

Common mistakes

Where Slope-Intercept Form usually goes wrong

The mistake	Why it is wrong	The fix
Writing $y = bx + m$ instead of $y = mx + b$ , swapping slope and y-intercept.	This reversal swaps which number multiplies $x$ and which is constant, causing the line to have the wrong slope and cross the y-axis at the wrong point.	Remember that $m$ (slope) always multiplies $x$ , and $b$ (y-intercept) is the constant term at the end; the order $mx + b$ is mandatory.
Reading $b = 2$ from $y = 3x + 2$ and plotting the point $(2, 0)$ instead of $(0, 2)$ on the y-axis.	The y-intercept $b$ is where the line crosses the y-axis (when $x = 0$ ), not the x-axis, so confusing axes leads to plotting the wrong starting point entirely.	Always plot the y-intercept at $(0, b)$ on the y-axis; set $x = 0$ to find it, not $y = 0$ .
Assuming a line cannot be written in slope-intercept form if the given information does not include the slope and y-intercept directly.	Any non-vertical line has a unique slope-intercept form; you can calculate the slope from any two points and find $b$ by substituting into the equation.	Always use the slope formula to calculate $m$ from two points, then substitute any point into $y = mx + b$ to solve for $b$ ; the form works for every non-vertical line.

Tips and when to use something else

If you're given a point and a slope, slope-intercept form is faster than Point-Slope Form because you can write the equation directly without rearranging.
When graphing, start at the y-intercept $(0, b)$ , then move by the slope: rise (up/down) by the numerator and run (left/right) by the denominator.
For a vertical line (where slope is undefined), slope-intercept form does not work; instead, write the equation as $x = c$ for the constant x-coordinate.
Always verify your final equation by substituting the y-intercept $(0, b)$ to confirm it gives $y = b$ .

Related formulas

Point-Slope FormPoint-Slope Form lets you write a line equation when you know the slope and one point; use it to quickly write equations or convert to standard form.Slope FormulaThe Slope Formula measures a line's steepness by dividing the vertical change between two points by their horizontal change.Parallel and Perpendicular SlopesFind whether two lines are parallel or perpendicular by comparing their slopes: identical for parallel, negative reciprocals for perpendicular.Standard Form of a LineStandard form Ax + By = C expresses a line's equation using integer coefficients, making it ideal for finding intercepts and solving systems of equations.Graphing Linear InequalitiesGraphing linear inequalities shows which coordinate points satisfy a condition like y > mx + b by shading solution regions with appropriate line styles.

Frequently asked questions

How do I find the slope and y-intercept if I only have two points?: Use the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ to find the slope. Then substitute either point into $y = mx + b$ and solve for $b$ . If one of the points is on the y-axis (where $x = 0$ ), that point's y-coordinate is $b$ directly.
What does it mean if the slope is 0?: A slope of 0 means the line is horizontal and does not rise or fall as $x$ changes. The equation becomes $y = b$ , a constant; every point on the line has the same y-value.
Can a line have a negative y-intercept?: Yes, absolutely. A negative $b$ means the line crosses the y-axis below the origin, at the point $(0, b)$ where $b$ is negative. This is perfectly valid and very common in real-world situations.
If I have the equation $3x - y = 6$ , how do I rewrite it as slope-intercept form?: Solve for $y$ : subtract $3x$ from both sides to get $-y = -3x + 6$ , then divide every term by $-1$ to get $y = 3x - 6$ . Now you can see directly that $m = 3$ and $b = -6$ .

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Reviewed 2026-09-18