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

Distribute the slope $m$ across the parentheses on the right side, then add or subtract $y_1$ from both sides to isolate $y$. The result will be in the form $y = mx + b$, where $b$ is the y-intercept.

Can you use any point on the line for point-slope form?

Yes. If a line passes through multiple known points, you can use any one of them with the slope. The resulting equations will look different but are algebraically equivalent — they all simplify to the same slope-intercept form.

When should you use point-slope form instead of slope-intercept form?

Use point-slope form whenever you know the slope and a point that is not the y-intercept. It is also the most efficient form when you are given two points, because you can plug in immediately after computing the slope. Slope-intercept form is more convenient when you already know the y-intercept.

Point-Slope Form — Definition, Formula & Examples

Point-slope form is a way to write the equation of a line when you know its slope and one point on the line. It is written as

y - y_1 = m(x - x_1)

, where

m

is the slope and

(x_1, y_1)

is the known point.

Point-slope form is a linear equation format derived from the definition of slope. Given a line with slope $m$ passing through the point $(x_1, y_1)$ , every other point $(x, y)$ on the line satisfies the equation $y - y_1 = m(x - x_1)$ . This form is algebraically equivalent to slope-intercept form and standard form, and it can be converted to either by simplifying and rearranging terms.

Key Formula

y - y_1 = m(x - x_1)

Where:

$m$ = Slope of the line (rise over run)
$(x_1, y_1)$ = A known point on the line
$(x, y)$ = Any other point on the line (left as variables)

How It Works

To use point-slope form, you need two pieces of information: a slope

m

and a single point

(x_1, y_1)

on the line. Substitute these values directly into the template

y - y_1 = m(x - x_1)

. Pay careful attention to the subtraction signs — if your point has a negative coordinate, subtracting a negative becomes addition. Once you have the equation, you can leave it in point-slope form or distribute and simplify to convert it into slope-intercept form (

y = mx + b

). If you are given two points instead of a slope, first calculate the slope using

m = \frac{y_2 - y_1}{x_2 - x_1}

, then pick either point to plug in.

Worked Example

Problem: Write the equation of a line with slope 3 that passes through the point (2, 5).

Step 1: Identify the slope and the point. Here

m = 3

and

(x_1, y_1) = (2, 5)

Step 2: Substitute into the point-slope formula.

y - 5 = 3(x - 2)

Step 3: This is the equation in point-slope form. To convert to slope-intercept form, distribute the 3.

y - 5 = 3x - 6

Step 4: Add 5 to both sides to isolate

y

y = 3x - 1

Answer: Point-slope form:

y - 5 = 3(x - 2)

. Slope-intercept form:

y = 3x - 1

Another Example

This example starts from two points instead of a given slope, and involves a negative coordinate, showing how double negatives simplify.

Problem: Write the equation of the line passing through the points (−1, 4) and (3, −8).

Step 1: Calculate the slope from the two points.

m = \frac{-8 - 4}{3 - (-1)} = \frac{-12}{4} = -3

Step 2: Choose either point. Using

(x_1, y_1) = (-1, 4)

, substitute into the formula.

y - 4 = -3(x - (-1))

Step 3: Simplify the double negative inside the parentheses.

y - 4 = -3(x + 1)

Step 4: Optionally convert to slope-intercept form by distributing and adding 4.

y - 4 = -3x - 3 \implies y = -3x + 1

Answer: Point-slope form:

y - 4 = -3(x + 1)

. Slope-intercept form:

y = -3x + 1

Visualization

Why It Matters

Point-slope form appears constantly in Algebra 1, Algebra 2, and precalculus courses whenever you need to write a line through a specific point. In calculus, the tangent line to a curve at a point is written using point-slope form, since you know the point of tangency and the derivative gives the slope. Professionals in data science, physics, and engineering use it to build linear models from measured data points.

Common Mistakes

Mistake: Flipping the subtraction signs — writing

y + y_1 = m(x + x_1)

instead of using minus signs.

Correction: The formula always subtracts:

y - y_1

and

x - x_1

. If a coordinate is negative, subtracting a negative produces a plus sign naturally, e.g.,

x - (-3) = x + 3

Mistake: Using the slope formula result incorrectly by swapping which point's coordinates go where.

Correction: When computing

m = \frac{y_2 - y_1}{x_2 - x_1}

, make sure the same point's coordinates are in the same position (both first or both second). The order of the points does not matter, but it must be consistent in numerator and denominator.

Mistake: Forgetting to distribute the slope to both terms inside the parentheses when converting to slope-intercept form.

Correction: For example,

3(x - 2)

must become

3x - 6

, not

3x - 2

. Multiply the slope by every term inside the parentheses.

Check Your Understanding

Write the equation in point-slope form for a line with slope

-2

passing through

(4, 1)

Hint: Plug

m = -2

x_1 = 4

, and

y_1 = 1

directly into

y - y_1 = m(x - x_1)

Answer:

y - 1 = -2(x - 4)

A line passes through

(0, -3)

and

(6, 9)

. Write its equation in point-slope form, then convert to slope-intercept form.

Hint: First find the slope with the slope formula. Then choose either point.

Answer: Slope:

m = 2

. Point-slope:

y + 3 = 2(x - 0)

. Slope-intercept:

y = 2x - 3

The equation

y + 7 = 5(x - 3)

is in point-slope form. What are the slope and the point used?

Hint: Rewrite

y + 7

y - (-7)

to identify

y_1

Answer: Slope is

5

; the point is

(3, -7)

Related Terms

Equation of a Line — Parent topic covering all line equation forms
Linear Equation — General category that includes point-slope form
Linear — Describes the type of relationship point-slope form represents
Horizontal Line Equation — Special case where slope is zero
Line of Best Fit — Uses point-slope form to model data trends
Cross Multiplication — Technique sometimes used when solving linear equations
Linear Combination — Method for solving systems of linear equations
Consistent System of Equations — System where lines written in point-slope form intersect