Point Slope Form Calculator
Create a line equation from a known point and slope, with equivalent point-slope, slope-intercept, and standard forms.
Find a line equation
Enter one point on the line and its slope.
About point-slope form
Point-slope form describes a nonvertical straight line when one point and its slope are known. The formula is y minus y1 equals m times x minus x1, where m is the slope and the ordered pair x1, y1 is any point on the line. It comes directly from the slope definition: the change in y divided by the change in x equals m. Multiplying both sides by the horizontal change gives the familiar point-slope equation.
This form is especially convenient because it requires no search for the y-intercept. If a problem says that a line has slope 4 and passes through the point 2, 3, substitute those three values to obtain y minus 3 equals 4 times x minus 2. Every coordinate pair satisfying that equation lies on the same line. Substituting x equal to 2 immediately returns y equal to 3, confirming that the specified point is included.
The calculator also converts the result to slope-intercept form. Expanding the right side and isolating y produces y equals mx plus b, where b equals y1 minus m times x1. In the example, b is 3 minus 8, or negative 5, so the equation is y equals 4x minus 5. This equivalent form makes the vertical intercept easy to read and is useful for graphing. A displayed standard form moves the x and y terms to one side, providing another common representation for algebra work.
Slope records direction and steepness. A positive slope rises from left to right, a negative slope falls, and a zero slope creates a horizontal line. Fractional and decimal slopes are accepted, so a slope of 0.5 means that y rises one unit for every two units of horizontal movement. A vertical line cannot be represented by a finite numerical slope because its horizontal change is zero. Such a line is instead written x equals a constant and is outside this calculator's point-slope input model.
Equivalent equations can look different after signs are simplified or both sides are multiplied by a nonzero constant. For example, negative factors may be distributed or left outside parentheses without changing the solution set. Use the results to check algebra, convert textbook exercises, prepare a line for graphing, or verify an equation derived from data. The substitutions are shown compactly so the relationship among the point, slope, and three equation forms remains clear.
Point-slope form examples
| Point and slope | Point-slope equation | Slope-intercept form |
|---|---|---|
| Point (2, 3), slope 4 | y - 3 = 4(x - 2) | y = 4x - 5 |
| Point (-1, 2), slope -3 | y - 2 = -3(x + 1) | y = -3x - 1 |
| Point (5, -2), slope 0 | y + 2 = 0(x - 5) | y = -2 |
How to use the point-slope calculator
- Enter the x-coordinate of a known point on the line.
- Enter the matching y-coordinate and the line's slope.
- Select Find line equation to substitute the values into point-slope form.
- Read the equivalent slope-intercept and standard forms below the primary result.
Point-slope form FAQ
What is the point-slope formula?
The formula is y minus y1 equals m times x minus x1. The pair x1, y1 is a known point and m is the line's slope.
How do I convert point-slope form to slope-intercept form?
Distribute the slope across the parentheses and then isolate y. The intercept can also be calculated directly as b equals y1 minus m times x1.
Can I use a negative slope?
Yes, negative, positive, zero, fractional, and decimal slopes are valid. A negative slope indicates that the line falls as x increases.
Can this calculator represent a vertical line?
No, a vertical line has undefined slope because its horizontal change is zero. Write a vertical line through x1 directly as x equals x1.
Does the chosen point have to be the y-intercept?
No, point-slope form works with any known point on the line. That flexibility is its main advantage over starting with slope-intercept form.