Point Slope Form Calculator

A point slope form calculator turns one point and a slope into the equation of a line. The form is y − y₁ = m(x − x₁). For the point (2, −5) with slope 3/4 the answer is y + 5 = (3/4)(x − 2), which rearranges to y = (3/4)x − 13/2 and to the standard form 3x − 4y = 26.

Three sister tools cover the neighboring questions. The Slope Calculator finds m on its own from two points when the equation is not what you need yet, the Linear Equation Calculator solves an equation for x once you already have one, and the Midpoint Calculator and Distance Calculator answer the other two standard questions about a pair of points. Use this page when the assignment says "write the equation of the line".

The point-slope formula

Point-slope form is nothing more than the slope definition rearranged. Slope between a fixed point (x₁, y₁) and any other point (x, y) on the same line is m = (y − y₁) ÷ (x − x₁). Multiply both sides by (x − x₁) and the fraction disappears:

That is the whole method. There is no solving, no isolating, no guessing: you substitute the slope for m and the coordinates of your point for x₁ and y₁, and the line is written. Because the subtraction signs are baked into the formula, a negative coordinate flips to a plus sign in the final equation. With x₁ = −3 the bracket becomes (x + 3), and with y₁ = −5 the left side becomes y + 5.

When you start from two points instead, you need one extra step first. The slope comes from the rise over the run:

Then you pick either point, put it into the point-slope formula, and you are finished. The calculator above does both routes, keeps every value as an exact fraction, and shows the algebra line by line.

Worked example: a point and a slope

Write the equation of the line through (2, −5) with slope m = 3/4.

1. Substitute. y − (−5) = (3/4)(x − 2), which tidies to y + 5 = (3/4)(x − 2). That is the point-slope answer.
2. Distribute the slope. (3/4)(x − 2) = (3/4)x − 3/2, so y + 5 = (3/4)x − 3/2.
3. Isolate y. Subtract 5 from both sides: y = (3/4)x − 3/2 − 5. Since 5 = 10/2, the constant is −3/2 − 10/2 = −13/2, giving y = (3/4)x − 13/2, or y = 0.75x − 6.5 in decimals.
4. Clear the fractions. Multiply everything by 4: 4y = 3x − 26. Move the x-term left: 3x − 4y = 26, the standard form with whole-number coefficients.
5. Check. At x = 2 the slope-intercept form gives 0.75 × 2 − 6.5 = 1.5 − 6.5 = −5, exactly the y-coordinate you started with.

The finished line crosses the y-axis at (0, −6.5) and the x-axis at (26/3, 0), which is about (8.67, 0). Its angle with the x-axis is arctan(0.75) = 36.87 degrees, and as a percent grade the slope reads 75%. Any line perpendicular to it has slope −4/3, the negative reciprocal.

Worked example: two points

Write the equation of the line through (−3, 4) and (5, −2).

1. Slope. The rise is −2 − 4 = −6 and the run is 5 − (−3) = 8, so m = −6/8 = −3/4. The negative sign says the line falls from left to right.
2. Point-slope. Using the first point: y − 4 = (−3/4)(x + 3). Using the second point instead gives y + 2 = (−3/4)(x − 5). Both are correct.
3. Slope-intercept. Distributing the first version gives y − 4 = (−3/4)x − 9/4, so y = (−3/4)x − 9/4 + 4 = (−3/4)x + 7/4, that is y = −0.75x + 1.75.
4. Standard form. Multiply by 4: 4y = −3x + 7, then move the x-term across to get 3x + 4y = 7.
5. Check both points. At x = −3: −0.75 × (−3) + 1.75 = 2.25 + 1.75 = 4. At x = 5: −0.75 × 5 + 1.75 = −3.75 + 1.75 = −2. Both original points sit on the line.

Two extra facts about the same pair of points: their midpoint is (1, 1) and the distance between them is exactly 10 units, because the rise of 6 and the run of 8 form a 6-8-10 right triangle. That is a useful sanity check, since a slope of −3/4 is the same 3-4 ratio in reduced form.

Converting point-slope form to slope-intercept form

Most homework asks for the conversion, and it is two moves. First distribute the slope across the bracket. Second, add or subtract the constant so that y stands alone on the left. In symbols, y − y₁ = m(x − x₁) becomes y = mx + (y₁ − m · x₁), which means the y-intercept has a shortcut of its own:

For (2, −5) and m = 3/4 that shortcut gives b = −5 − (3/4)(2) = −5 − 1.5 = −6.5 in a single line, matching the long route above. Use it when you only want the intercept and do not need the intermediate algebra written out.

Converting to standard form Ax + By = C

Standard form wants integer coefficients, the variables on the left, the constant on the right, and by convention A greater than or equal to zero. Getting there from slope-intercept form takes three moves: multiply every term by the least common denominator to clear fractions, move the x-term to the left side, and if the x-coefficient came out negative, multiply the whole equation by −1. Finally divide by any common factor so the three numbers share no divisor. The calculator applies exactly that sequence, which is why y = (3/4)x − 13/2 becomes 3x − 4y = 26 rather than the equivalent but untidy 6x − 8y = 52.

Point and slope converted into all three forms

Every row below was produced with the formulas on this page. Notice how a negative x₁ or y₁ always turns into a plus sign inside the finished equation.

Two points converted into a line equation

Here the slope is computed first with the rise-over-run formula, and the first listed point is the one substituted into point-slope form. The last row is the exception that has no point-slope version at all.

What each slope looks like

The slope you type controls the steepness and the direction of the line. The angle column is arctan(m) converted to degrees, and the percent grade is simply the slope times 100, the same convention road signs use.

A slope of 1 is the benchmark worth memorizing: rise equals run, so the line tilts at exactly 45 degrees. Anything steeper has a slope above 1, anything gentler sits between 0 and 1.

How to use this point slope calculator

1. Pick your starting information. Tap "A point and the slope" if the problem hands you m directly, or "Two points" if you have to work the slope out first.
2. Type the coordinates. Enter x₁ and y₁ for your point. Negative values are fine, and so are decimals such as −1.5.
3. Enter the slope as a fraction if you have one. Typing 3/4 keeps the answer exact; typing 0.75 gives the same line but shows the constants as decimals wherever the fraction does not terminate neatly.
4. Read the headline. The blue card shows the finished point-slope equation, with slope-intercept form, standard form, m and b in the tiles beneath it.
5. Follow the steps. The step-by-step card writes out the substitution, the distribution, the value of b and the rearrangement into standard form, so you can copy the reasoning rather than just the answer.
6. Use the table of points. The last card lists six coordinate pairs on the line, which is all you need to sketch it on graph paper.

Who this calculator is for

• Algebra 1 and Algebra 2 students working through a homework set on writing linear equations.
• Students revising for a test who want a worked solution to compare against their own, not just a final answer.
• Parents and tutors checking a page of exercises quickly without redoing every conversion by hand.
• Anyone fitting a straight line to two data readings, such as a cost that grows by a fixed amount per unit, and needing the formula behind it.
• Trades and design work where a run and a rise are known and the equation of the line matters for layout.

Key terms explained

• Slope (m): how much y changes for each one-unit increase in x. Positive rises, negative falls, zero is flat.
• Rise and run: the vertical and horizontal change between two points. Slope is rise divided by run.
• Point-slope form: y − y₁ = m(x − x₁), built from any single point on the line.
• Slope-intercept form: y = mx + b, where b is the y-value at x = 0.
• Standard form: Ax + By = C with integer coefficients, the version most often used when solving systems of equations.
• y-intercept: the point (0, b) where the line crosses the vertical axis.
• x-intercept: the point where y = 0, found by solving 0 = mx + b, so x = −b/m.
• Negative reciprocal: the slope of a perpendicular line. Flip the fraction and change the sign, so 3/4 becomes −4/3.

Special cases: horizontal and vertical lines

A horizontal line has slope 0. Point-slope form still works, it just collapses: y − 7 = 0(x − 3) is true for every x, which means y = 7 everywhere. A vertical line is the one case point-slope form cannot express. Both of its points share the same x-value, so the run is zero, and slope is rise divided by run. Division by zero is undefined, so there is no m to substitute. Write the line as x = 7 instead, or as the standard form 1x + 0y = 7. The calculator detects this situation in two-point mode and tells you so rather than printing an error.

What changes the result

• The slope sets the steepness and the direction. Change its sign and the line mirrors from rising to falling.
• The point shifts the line up or down without tilting it. Two lines with the same m but different points are parallel.
• Which point you choose changes how the point-slope equation looks, but never the line itself. Both versions simplify identically.
• Fractions versus decimals. A slope of 1/3 is exact, while 0.333 is not; over a long run that rounding visibly moves the line.
• Sign errors on negative coordinates are the single most common way a correct slope still produces the wrong equation.

Tips that save time

• When both points have ugly coordinates, use point-slope form and stop there if the question does not demand slope-intercept form. It is a complete, valid equation.
• Keep the slope as a fraction until the very last step, then convert to a decimal only if the answer sheet asks for one.
• Use b = y₁ − m × x₁ as a one-line shortcut whenever you only need the y-intercept.
• For a parallel line through a new point, reuse the same m. For a perpendicular one, flip the fraction and change the sign.
• Always test the original point in your final equation. It catches nearly every sign slip in a few seconds.
• If the run divides the rise evenly, the slope is a whole number and the algebra stays fraction-free the whole way.

Limitations of this tool

• It handles straight lines only. Parabolas, circles and other curves need a different method; for a quadratic, use the Quadratic Formula Calculator.
• It works in two dimensions. A line in three-dimensional space needs a vector or parametric form instead.
• Vertical lines have no point-slope form, so the calculator reports x = c rather than forcing an equation that does not exist.
• Very long decimal inputs are converted to exact fractions, which can produce large numerators in standard form. Rounding your input first keeps the coefficients readable.
• It does not draw a plotted graph. The table of points is there so you can sketch the line yourself.

Which calculator should you use?

Line problems come in a small family, and each sibling page answers a different part of it:

• Use this page when you need the equation of a line from a point and a slope, or from two points.
• Use the Slope Calculator when the question stops at m, the rise over run, the angle or the distance between two points.
• Use the Linear Equation Calculator when you already have an equation and need to solve it for x.
• Use the Midpoint Calculator for the point exactly halfway between two coordinates, and the Distance Calculator for the length of the segment joining them.
• Use the Inequality Calculator when the relation is a less-than or greater-than rather than an equals sign.

Sources

Everything on this page is deterministic coordinate geometry, so no external data source applies. The point-slope identity y − y₁ = m(x − x₁) follows algebraically from the definition of slope, m = (y₂ − y₁) ÷ (x₂ − x₁); the slope-intercept and standard forms are equivalent rearrangements of that same equation, and the angle of the line is arctan(m) by the definition of the tangent function. These are exact mathematical definitions rather than measured or published values, so there is nothing to cite and nothing that changes over time. The only convention involved is the standard-form tidying rule, that A, B and C are integers with no common factor and A is not negative, which is the ordering used in US algebra curricula.