What Is Point-Slope Form?
Point-slope form is a way to write the equation of a straight line when you know one point on the line and its slope. The formula is y - y₁ = m(x - x₁), where (x₁, y₁) is the known point and m is the slope. This form is especially useful because it directly reflects the geometric definition of a line: starting from a known point, every other point on the line is reached by moving along the slope.
Engineers and scientists use point-slope form frequently because measurements often provide a single data point and a rate of change. For example, if a car is at mile marker 50 and traveling at 60 miles per hour, the point-slope form immediately gives you its position at any time. For calculating slope from two points, see our Slope Calculator.
What This Calculator Does
This point-slope calculator takes a single point and a slope, then converts them into three common forms of a linear equation: point-slope form, slope-intercept form, and standard form. It also computes the y-intercept.
- Inputs: One point (x₁, y₁) and the slope m
- Outputs: Point-slope form, slope-intercept form, standard form, and y-intercept
How the Calculation Works
Point-Slope: y - y₁ = m(x - x₁)
Slope-Intercept: y = mx + b, where b = y₁ - m·x₁
Standard Form: Ax + By = C
Starting from the point-slope form, you expand and rearrange to get the other forms. Distributing m gives y - y₁ = mx - mx₁, then adding y₁ to both sides produces y = mx + (y₁ - mx₁), which is slope-intercept form with b = y₁ - mx₁. To convert to standard form, move the x term to the left side: -mx + y = b, then normalize so that the coefficient of y is positive. For solving quadratic relationships, try our Quadratic Equation Calculator.
Example Calculation
Example: Given the point (2, 3) and a slope of 2, the point-slope form is y - 3 = 2(x - 2). Expanding gives y - 3 = 2x - 4, so y = 2x - 1 in slope-intercept form. The y-intercept is -1. Converting to standard form: 2x - y = 1. You can verify that the point (2, 3) satisfies all three forms.
When to Use Each Form
Point-slope form is best when you have a point and a slope and want to write the equation quickly. Slope-intercept form (y = mx + b) is ideal for graphing because it directly shows the slope and y-intercept. Standard form (Ax + By = C) is preferred in systems of equations and certain standardized tests because it makes elimination and comparison easier. For distance calculations between points, use our Distance Calculator.