Point-Slope Form Calculator (From a Point and Slope or Two Points, with Steps)

To write a line in point-slope form, put one point (x₁, y₁) and the slope m into y − y₁ = m(x − x₁), and this point-slope form calculator does that for you from a point and a slope or from two points. It also gives the slope-intercept and standard forms with exact fractions, shows every step, and graphs the line, so you can check your own work line by line.

How to use the Point-Slope Form Calculator

1. Choose A point and the slope or Two points.
2. Type the coordinates and, if you have it, the slope. Integers, decimals and fractions such as -5/3 all work. For a vertical line, type undefined as the slope.
3. Press Calculate or the Enter key.
4. Read the point-slope, slope-intercept and standard form answers, then follow the numbered steps and the graph to see where each number came from.
5. Scroll down to the practice questions to try the same skills on your own.

How to write point-slope form by hand

Point-slope form is y − y₁ = m(x − x₁). The letter m is the slope, and (x₁, y₁) is any point on the line. The form comes straight from the slope formula: if (x, y) is any other point on the line, then (y − y₁) ÷ (x − x₁) = m, and multiplying both sides by (x − x₁) gives the equation.

Example 1: a point and a slope

Write the equation of the line through (3, −2) with slope 4.

1. Substitute x₁ = 3, y₁ = −2 and m = 4: y − (−2) = 4(x − 3).
2. Clean up the double negative: y + 2 = 4(x − 3). That is the point-slope form.
3. For slope-intercept form, distribute: y + 2 = 4x − 12. Subtract 2 from both sides: y = 4x − 14.
4. For standard form, move the x-term to the left: −4x + y = −14. Multiply by −1 so A is positive: 4x − y = 14.
5. Check with the original point: 4(3) − (−2) = 12 + 2 = 14. It works.

Example 2: two points

Write the equation of the line through (−2, 1) and (4, 5).

1. Find the slope: m = (5 − 1) ÷ (4 − (−2)) = 4 ÷ 6 = 2/3.
2. Use the first point: y − 1 = 2/3(x + 2).
3. Distribute: y − 1 = 2/3 x + 4/3. Add 1, which is 3/3: y = 2/3 x + 7/3.
4. Standard form: −2/3 x + y = 7/3. Multiply every term by 3 to clear the fractions: −2x + 3y = 7. Make A positive: 2x − 3y = −7.
5. Check with the second point: 2(4) − 3(5) = 8 − 15 = −7. It works.

If you start from the other point instead, you get y − 5 = 2/3(x − 4). It looks different, but distributing gives y = 2/3 x − 8/3 + 5 = 2/3 x + 7/3, the same line. Any point on the line is a valid choice.

If the slope step is where you get stuck, the slope calculator walks through rise over run on its own, and the simplify fractions calculator helps with reducing answers like 4/6.

Quick reference: forms of a linear equation

Common mistakes to avoid

• Sign errors with negative coordinates. The form has minus signs built in, so a point like (−5, −1) gives y + 1 = m(x + 5), not y − 1 = m(x − 5).
• Mixing up the order in the slope formula. If you subtract y₂ − y₁ on top, you must subtract x₂ − x₁ on the bottom. Reversing only one of them flips the sign of the slope.
• Putting rise and run upside down. Slope is the change in y divided by the change in x.
• Taking x₁ from one point and y₁ from the other. Both coordinates must come from the same point.
• Distributing to only one term. In 2/3(x + 2), the 2/3 multiplies both x and 2.
• Forcing a vertical line into point-slope form. When x₁ = x₂, the run is 0 and the slope is undefined, so the answer is simply x = a.

Where point-slope form fits in the curriculum

Students usually meet point-slope form in Algebra 1, right after slope-intercept form. It builds on Common Core standard 8.F.B.4, which asks students to find the rate of change and initial value of a linear function from two (x, y) values, and on 8.EE.B.6, which connects slope to the equations y = mx and y = mx + b. In high school it supports A-CED.A.2, creating equations in two variables to represent relationships.

It is the most direct form when a problem hands you a point and a rate of change. A taxi that charges $2.50 per mile and costs $13 for a 4-mile ride gives the point (4, 13) and slope 2.5, so y − 13 = 2.5(x − 4) right away. Later, in calculus, the equation of a tangent line is written in exactly this form.

Check your answer

Plug each given point into your final equation. Both sides should match. With two points, test the point you did not use to write the equation, because that is the one that catches slope mistakes. You can also compare slopes: in slope-intercept form, the number in front of x must equal the slope you found. For more practice finding slopes first, try the slope practice questions.

Point-slope form practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

What is point-slope form?

Point-slope form is a way to write the equation of a straight line: y − y1 = m(x − x1), where m is the slope and (x1, y1) is a known point on the line. It is useful because you can write it as soon as you know one point and the slope.

How do you write point-slope form from two points?

First find the slope with m = (y2 − y1) ÷ (x2 − x1). Then substitute that slope and either one of the two points into y − y1 = m(x − x1). For (−2, 1) and (4, 5), the slope is 2/3 and one answer is y − 1 = 2/3(x + 2).

Does it matter which point I use?

No. Different points give equations that look different, but they describe the same line. If you simplify both to slope-intercept form, you get the same y = mx + b. Teachers usually accept either version unless the problem names a point.

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

Distribute the slope on the right side, then move the constant from the left side to the right. For y + 2 = 4(x − 3), distributing gives y + 2 = 4x − 12, and subtracting 2 gives y = 4x − 14.

Can a vertical line be written in point-slope form?

No. A vertical line has an undefined slope, so there is no number to put in place of m. Its equation is x = a, where a is the x-coordinate shared by every point on the line. A horizontal line is fine: its slope is 0, so y − y1 = 0(x − x1), which simplifies to y = y1.

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

Both describe the same line. Point-slope form uses any point on the line, while slope-intercept form, y = mx + b, uses the specific point where the line crosses the y-axis. Point-slope form is faster to write; slope-intercept form is easier to graph and compare.