Find the equation of a straight line in slope-intercept form (y = mx + b), point-slope form, and standard form. Enter two points, or give a point and a slope. The calculator shows full step-by-step working.
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Standard formula
Coordinate geometry: slope formula and slope-intercept form.
Step 2 - Y-intercept: b = y1 - m * x1 = 3 - 2 * 1 = 1
Slope-intercept form: y = 2x + 1
Point-slope form: y - 3 = 2(x - 1)
Standard form: 2x - y = -1
How to Find the Equation of a Line
A straight line in a two-dimensional plane is described completely by its slope (gradient) and any one point through which it passes. From these two pieces of information, you can express the line in any standard form. The three most common forms are:
Form
Equation
When to use
Slope-intercept
y = mx + b
When you know or need the slope and y-intercept directly
Point-slope
y - y1 = m(x - x1)
When you have a specific point and the slope, before finding b
Standard form
Ax + By = C
For comparing lines, or when integer coefficients are preferred
Finding the Equation from Two Points
Given two points (x1, y1) and (x2, y2), follow these steps:
Calculate the slope: m = (y2 - y1) / (x2 - x1). The slope tells you how much y changes for every 1 unit increase in x.
Find the y-intercept: Substitute the slope and one point into y = mx + b and solve for b: b = y1 - m * x1.
Write the equation: Substitute m and b into y = mx + b.
Vertical lines: When two points share the same x-coordinate, the slope is undefined (division by zero). The line is expressed as x = constant, for example x = 3. Vertical lines have no y-intercept unless they pass through the origin.
Horizontal lines: When two points share the same y-coordinate, the slope is 0. The equation is simply y = b, where b is the shared y-value. For example, through (2, 5) and (7, 5): y = 5.
Converting Between Forms
Starting from slope-intercept form y = mx + b, you can convert to standard form by moving all terms to one side: mx - y = -b. To ensure the coefficient of x is positive, multiply through by -1 if needed: -mx + y = b, or equivalently Ax + By = C.
To convert from standard form Ax + By = C back to slope-intercept, solve for y: y = (-A/B)x + C/B.
Using the X-intercept
The x-intercept is the point where the line crosses the x-axis (where y = 0). Set y = 0 in the slope-intercept equation: 0 = mx + b, so x = -b/m. For y = 2x + 1, the x-intercept is x = -1/2 = -0.5.
Sources and method: Coordinate geometry: slope formula m = (y2 - y1) / (x2 - x1); slope-intercept form y = mx + b; point-slope form y - y1 = m(x - x1); standard form Ax + By = C. Standard algebra textbook derivations.
This calculator works with real numbers. Results are rounded to 6 significant figures for display. For lines expressed as fractions, the exact decimal approximation is shown. Vertical lines (undefined slope) are noted but cannot be expressed in slope-intercept or point-slope form.
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