Point-slope form of a line

Point-slope form is a way to write the equation of a straight line when you know its slope and one point on it: y − y₁ = m(x − x₁). Here m is the slope and (x₁, y₁) is the point. Put in the numbers and you have the equation straight away, with no need to find the y-intercept first.

Point-slope form formula

The small 1s are labels, not exponents. They mark the coordinates of the specific point you know, while plain x and y stay in the final equation.

Where the formula comes from

Slope is the change in y divided by the change in x between any two points. Take your known point (x₁, y₁) and any other point (x, y) on the line:

Multiply both sides by (x − x₁) and you get y − y₁ = m(x − x₁). So point-slope form is just the slope formula, rearranged.

How to write an equation in point-slope form

1. Find the slope m. It may be given, or you may need to work it out from two points.
2. Pick one point on the line and call it (x₁, y₁).
3. Substitute m, x₁ and y₁ into y − y₁ = m(x − x₁). Leave x and y as letters.
4. Tidy the signs. Subtracting a negative becomes adding: y − (−3) is y + 3.

Example 1: a slope and a point

Write the equation of the line with slope 2 through (3, 5).

That is the answer in point-slope form. Nothing else is needed unless the question asks for another form.

Example 2: negative coordinates

Write the equation of the line with slope −4 through (−2, 7).

Watch the signs: x minus negative 2 becomes x + 2. The point is still (−2, 7).

Point-slope form from two points

Given two points, find the slope first, then use either point.

Example 3: through (1, −3) and (5, 9)

Using (1, −3) as the point:

Using (5, 9) instead gives y − 9 = 3(x − 5). Both are correct. They look different, but they describe the same line, and both simplify to y = 3x − 6.

Here x and y are the coordinates. Both points sit on the line, which rises 3 for every 1 across: the slope. You can plot your own lines in the Linear Graph calculator.

Finding the slope from two points is the step most mistakes happen in. Try one here, then press Mark.

Point-slope form to slope-intercept form

Slope-intercept form is y = mx + b, where b is the y-intercept. To convert, distribute the slope and get y by itself.

Example 4: convert y − 7 = −4(x + 2)

The slope is −4 and the y-intercept is −1. Check with the original point: −4(−2) − 1 = 7, which matches (−2, 7).

Point-slope form to standard form

Standard form is Ax + By = C, usually with whole numbers and A positive. Clear any fractions, then move the x and y terms to one side.

Example 5: a fractional slope

Write the line with slope 2/3 through (−3, 4) in point-slope, slope-intercept and standard form.

For standard form, multiply the slope-intercept form by 3 to clear the fraction, then rearrange:

Check with the point: 2(−3) − 3(4) = −6 − 12 = −18. It works.

Which form should you use?

Reading and graphing point-slope form

You can read the point and slope straight off an equation, but the point's signs are the opposite of what you see. In y + 2 = −3(x − 4), the point is (4, −2), because x − 4 = 0 when x = 4 and y + 2 = 0 when y = −2. The slope is −3.

1. Plot the point (4, −2).
2. From that point, use the slope as rise over run: −3 means down 3, right 1.
3. Plot the second point at (5, −5) and draw the line through both.

Parallel and perpendicular lines

Point-slope form is the fastest way to write a line parallel or perpendicular to another one through a given point.

• Parallel lines have equal slopes. The line parallel to y = 5x − 2 through (2, −1) is y + 1 = 5(x − 2), which simplifies to y = 5x − 11.
• Perpendicular lines have slopes that multiply to −1. Flip the slope and change its sign: 5 becomes −1/5. The line perpendicular to y = 5x − 2 through (10, 3) is y − 3 = −⅕(x − 10), which simplifies to y = −⅕x + 5.

Horizontal and vertical lines

A horizontal line has slope 0. Through (3, 5), point-slope form gives y − 5 = 0(x − 3), so y = 5. A vertical line has an undefined slope, so it cannot be written in point-slope form at all. The vertical line through (3, 5) is simply x = 3.

Using point-slope form in word problems

Example 6: a phone plan

A phone plan has a setup fee plus a fixed monthly charge. After 4 months you have paid230 in total, and after 10 months 470. Write an equation for the total cost y after x months, and find the setup fee.

The two facts are the points (4, 230) and (10, 470). The slope is the monthly charge:

The plan costs 40 a month</strong> and the setup fee, the cost at x = 0, is <strong>70.

Here x is the number of months and y is the total paid in dollars. Point-slope form let you write the equation from the two facts you had, without knowing the setup fee first.

Common mistakes

• Sign errors with negative coordinates. For the point (−2, 7), it is x + 2, not x − 2. Write x − (−2) first, then simplify.
• Swapping x and y. The y-coordinate goes with y on the left, and the x-coordinate goes with x inside the parentheses.
• Mixing the order in the slope formula. If you start with y₂ on top, start with x₂ on the bottom.
• Forgetting to distribute. −4(x + 2) is −4x − 8, not −4x + 2.
• Replacing x and y with numbers. Only x₁ and y₁ get numbers. Plain x and y stay as letters.
• Reading the point with the wrong signs. In y − 6 = 2(x + 5), the point is (−5, 6), not (5, −6).

Point-slope form practice problems

Write each line in point-slope form, then in slope-intercept form. Answers are below.

1. Slope −3 through (2, 4)
2. Through (0, 5) and (4, −3)
3. Slope ½ through (−4, 1)
4. Through (−1, −6) and (3, 2)
5. Parallel to y = 2x + 7 through (−5, 6)

Answers: 1) y − 4 = −3(x − 2), so y = −3x + 10. 2) Slope −2, so y − 5 = −2(x − 0), or y = −2x + 5. 3) y − 1 = ½(x + 4), so y = ½x + 3. 4) Slope 2, so y + 6 = 2(x + 1), or y = 2x − 4. 5) y − 6 = 2(x + 5), so y = 2x + 16.

Practice with a short timed paper

Writing equations of lines comes up all through Algebra 1, Geometry and the SAT. The daily Pre-Algebra & Algebra 1 and Geometry, Algebra 2 & SAT Math papers are short timed practice tests of 8 questions, with a new one every day. For more on slopes and lines, work through the Graphing Functions course.

Read next

Maths · GraphsSlope calculator: find the slope of a line from two pointsFree slope calculator: type two points to get the slope as an exact fraction, the line in y = mx + b form and every step, with worked examples.