Point-Slope Equation of a Line

Point-Slope Equation of a Line

y – y₁ = m(x – x₁), where m is the slope and (x₁, y₁) is a point on the line. Point-slope is the form used most often when finding the equation of a line.

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Point and Slope:
How to find the equation of a line
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Two Points:
How to find the equation of a line
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Key Formula

y - y_1 = m(x - x_1)

Where:

$m$ = The slope of the line (rise over run)
$(x_1, y_1)$ = A known point that lies on the line
$x$ = The independent variable
$y$ = The dependent variable

Worked Example

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

Step 1: Identify the known values. The slope is m = 3, and the point on the line is (x₁, y₁) = (2, 5).

m = 3, \quad x_1 = 2, \quad y_1 = 5

Step 2: Substitute these values into the point-slope formula.

y - 5 = 3(x - 2)

Step 3: This is your equation in point-slope form. If you need slope-intercept form, distribute and simplify.

y - 5 = 3x - 6

Step 4: Add 5 to both sides to isolate y.

y = 3x - 1

Answer: The equation in point-slope form is y − 5 = 3(x − 2), which simplifies to y = 3x − 1 in slope-intercept form.

Another Example

This example starts with two points instead of a given slope, so you must first compute the slope. It also shows that choosing either point in the formula produces the same line.

Problem: Find the equation of the line passing through the points (1, 2) and (4, 8).

Step 1: Calculate the slope using the two points.

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 2}{4 - 1} = \frac{6}{3} = 2

Step 2: Choose either point to plug into the formula. Using (1, 2):

y - 2 = 2(x - 1)

Step 3: You could verify by using the other point (4, 8) instead. Both give the same line.

y - 8 = 2(x - 4)

Step 4: Simplify either form to slope-intercept to confirm they match.

y - 2 = 2x - 2 \implies y = 2x

Step 5: Check the second version: y − 8 = 2x − 8 also gives y = 2x. Both forms represent the same line.

y - 8 = 2x - 8 \implies y = 2x

Answer: The equation is y − 2 = 2(x − 1), which simplifies to y = 2x.

Frequently Asked Questions

What is the difference between point-slope form and slope-intercept form?

Point-slope form is y − y₁ = m(x − x₁) and uses any point on the line. Slope-intercept form is y = mx + b and specifically uses the y-intercept (0, b). You can always convert point-slope form into slope-intercept form by distributing and solving for y. Point-slope is more flexible because you don't need to know the y-intercept.

When should you use point-slope form?

Use point-slope form whenever you know the slope and at least one point on the line. It is especially useful when you are given two points (compute the slope first, then apply the formula) or when the y-intercept is not easily found. Most textbook problems that ask you to "find the equation of a line" are quickest to solve in point-slope form.

Does it matter which point you use in the point-slope formula?

No. If you have two or more points on the line, you may substitute any one of them for (x₁, y₁). The resulting equations look different in point-slope form, but they all simplify to the same line when you convert to slope-intercept or standard form.

Point-Slope Form vs. Slope-Intercept Form

	Point-Slope Form	Slope-Intercept Form
Formula	y − y₁ = m(x − x₁)	y = mx + b
What you need	Slope and any point on the line	Slope and the y-intercept
Best used when	Given a point and a slope, or two points	Reading a graph or when b is already known
Graphing ease	Less immediate — must simplify first or plot the given point and use slope	Very direct — plot b on the y-axis, then use slope
Flexibility	Works with any known point	Requires the specific point (0, b)

Why It Matters

Point-slope form appears constantly in algebra courses, standardized tests like the SAT and ACT, and any setting where you need to quickly write a line's equation from given information. In calculus, the tangent line to a curve at a point is written using this exact formula, with the derivative supplying the slope. Mastering it also builds your ability to move fluently between different forms of a linear equation, a skill used throughout higher mathematics and science.

Common Mistakes

Mistake: Flipping the signs inside the parentheses. For example, writing y − 5 = m(x + 3) when the point is (3, 5).

Correction: The formula subtracts the coordinates: y − y₁ and x − x₁. If the point is (3, 5), you must write (x − 3), not (x + 3). A negative coordinate like (−3, 5) gives x − (−3) = x + 3.

Mistake: Forgetting to distribute the slope to both terms when converting to slope-intercept form. For instance, writing y = 3x − 2 + 5 from y − 5 = 3(x − 2) without properly distributing to get 3x − 6.

Correction: Always multiply the slope m by every term inside the parentheses: 3(x − 2) = 3x − 6, not 3x − 2. Then combine constants carefully.

Related Terms

Slope of a Line — The m value used in point-slope form
Slope-Intercept Equation of a Line — Alternate form using the y-intercept
Standard Form for the Equation of a Line — Another common form: Ax + By = C
Equation of a Line — General overview of all line equation forms
Point — The known coordinate pair (x₁, y₁)
Line — The geometric object being described
Two Intercept Form for the Equation of a Line — Form using x-intercept and y-intercept
Horizontal Line Equation — Special case where slope m = 0