Point-slope form is a way to write the equation of a line from its slope m and one point (x1, y1) on it: y − y1 = m(x − x1). It is the fastest form to use when you know a point and the slope, or two points. Multiplying it out gives slope-intercept form.
In Spanish
forma punto-pendiente
Also called
point-slope equation, point-slope formula, point-slope form of a line
Point-slope form is the slope formula with the fraction cleared. The numbers in it are the point you start from and the slope you travel by: from (2, 1), every run of 2 rises 3.
Formal definition and notation
Let a non-vertical line have slopem and pass through the fixed point (x1, y1). For any other point (x, y) on the line, the slope formula gives y − y1x − x1 = m. Multiplying both sides by x − x1 gives y − y1 = m(x − x1), which the point (x1, y1) also satisfies. Any point on the line can play the role of (x1, y1), so one line has many point-slope equations. A vertical line x = x1 has no slope, so it has no point-slope form.
Symbol
Read it as
Meaning
(x1, y1)
x sub 1, y sub 1
one specific, known point on the line
m
m
the slope of the line
(x, y)
x, y
any point on the line; these stay as variables in the equation
Key facts
One line, many equations: the line through (1, 3) and (3, 7) has slope 2, and both y − 3 = 2(x − 1) and y − 7 = 2(x − 3) describe it.
Read the point and slope straight off the equation: y + 4 = −3(x − 2) has slope −3 and passes through (2, −4).
To get slope-intercept form, distribute and solve for y: y − 1 = 2(x − 3) becomes y = 2x − 5.
From two points: find the slope first, then use either point.
In a word problem, point-slope form fits a known rate plus one known data point, such as a total cost after a given number of hours.
Point-slope formula
Point-slope formula
y − y1 = m(x − x1)
m
the slope
x1, y1
the coordinates of a known point on the line
x, y
variables for every point on the line
Watch the signs. For the point (−2, 5): y − 5 = m(x − (−2)), which is y − 5 = m(x + 2).
Try it: pick a point and a slope
Interactive
Drag either point. The readout gives the slope and the line’s equation; write the point-slope form with point A, then with point B, and check that both give the same slope-intercept form. The presets show special cases.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Point-Slope Form examples, step by step
Example 1: From a point and a slope
Warm-up
Write the equation of the line with slope −4 through (3, −1) in point-slope form.
Substitutem = −4, x1 = 3 and y1 = −1 into y − y1 = m(x − x1).
Find the equation of the line through (−2, 5) and (4, −7). Give it in point-slope form and in slope-intercept form.
Slope: m = −7 − 54 − (−2) = −126 = −2.
Use the point (−2, 5): y − 5 = −2(x + 2).
Distribute: y − 5 = −2x − 4.
Add 5: y = −2x + 1.
Answer: Point-slope: y − 5 = −2(x + 2). Slope-intercept: y = −2x + 1.
Example 3: A rate and one data point
Test-style
A plumber charges a fixed call fee plus 65 dollars per hour. A 3-hour job costs 245 dollars. Write an equation for the cost y of an x-hour job. What does a 5-hour job cost, and what is the call fee?
The rate is the slope, m = 65, and (3, 245) is a point: y − 245 = 65(x − 3).
For x = 5: y = 245 + 65(5 − 3) = 245 + 130 = 375.
The call fee is the cost at x = 0: y = 245 + 65(0 − 3) = 245 − 195 = 50.
Answer:y − 245 = 65(x − 3). A 5-hour job costs 375 dollars, and the call fee is 50 dollars.
Common mistakes
✗ MistakeUsing the point (−3, 5) and writing y − 5 = 2(x − 3).
✓ Insteady − 5 = 2(x − (−3)) = 2(x + 3).
The formula subtracts x1. Subtracting a negative number gives a plus sign.
✗ MistakeSwapping the coordinates: for (4, 1), writing y − 4 = m(x − 1).
✓ Insteady − 1 = m(x − 4).
The y-coordinate goes with y and the x-coordinate goes with x.
✗ MistakeDistributing only to x: y − 4 = 3(x − 2) becomes y = 3x − 2 + 4.
✓ Insteady − 4 = 3x − 6, so y = 3x − 2.
The slope multiplies both terms in the parentheses, x and −2.
Slope-intercept form, y = mx + b, uses the y-intercept as its point. Point-slope form can use any point. For the line through (3, 7) with slope 2: y − 7 = 2(x − 3), or y = 2x + 1.
Slope is a single number, the steepness m. Point-slope form is a whole equation that uses that number together with a point.
Quick check: 3 questions
Question 1 of 3
What point and slope does y − 6 = 4(x + 1) show?
Show answer and explanation
Answer: B.x + 1 = x − (−1), so x1 = −1, and y − 6 gives y1 = 6. The other choices miss the sign flip, flip both signs, or swap the coordinates.
Question 2 of 3
Which is an equation of the line through (2, −3) with slope 5?
Show answer and explanation
Answer: D.y − (−3) = 5(x − 2) simplifies to y + 3 = 5(x − 2). The first choice uses +3 for y1, the second flips the x sign too, and the third swaps the coordinates.
Question 3 of 3
Write y − 4 = −2(x − 3) in slope-intercept form.
Show answer and explanation
Answer: C. Distribute: y − 4 = −2x + 6. Add 4: y = −2x + 10. −2x − 2 drops the sign of 6, −2x + 7 multiplies only x, and −2x + 1 subtracts 4 instead of adding it.
Point-Slope Form: frequently asked questions
When should you use point-slope form?
Use it when you know the slope and a point that is not the y-intercept, or when you know two points. You can write the equation in one step without solving for b. If a question asks for slope-intercept form, start with point-slope form and then distribute and solve for y.
How do you convert point-slope form to slope-intercept form?
Distribute the slope over the parentheses, then move y1 to the right side. For y + 2 = 3(x − 4): distribute to get y + 2 = 3x − 12, then subtract 2 to get y = 3x − 14. The slope stays 3, and −14 is the y-intercept.
Does it matter which point you use?
No. Any point on the line works, and the equations will look different but describe the same line. Through (1, 3) and (3, 7), both y − 3 = 2(x − 1) and y − 7 = 2(x − 3) simplify to y = 2x + 1. Picking the point with smaller or positive numbers just makes the arithmetic easier.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 8.EE.B.6, CCSS HSA-CED.A.2
Cite this entry
Effortless Math Team. “Point-Slope Form.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/point-slope-form/
Flashcard
TermPoint-Slope Formy − y1 = m(x − x1)
DefinitionThe equation of a line from its slope and one point. Slope 2 through (1, 3): y − 3 = 2(x − 1).
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