Slope-Intercept Form Calculator

Convert between y=mx+b, standard form, two points & point-slope — with interactive graph

Enter coefficients for Ax + By = C

Conversion Formula Quick Reference

Standard → Slope-Int
Ax + By = C
m = −A/B | b = C/B

Slope-Int → Standard
y = mx + b
Multiply by LCD, keep A > 0

Two Points → Slope
m = (y₂−y₁) / (x₂−x₁)
b = y₁ − m·x₁

Parallel & Perpendicular
Parallel: m₂ = m₁
Perpendicular: m₂ = −1/m₁

What is Slope-Intercept Form?

Slope-intercept form is the equation y = mx + b, where m is the slope of the line and b is the y-intercept — the point where the line crosses the vertical axis. It is by far the most widely used way to express a linear equation in algebra, pre-calculus, and beyond, because both key properties of the line are immediately readable from the equation itself without any additional algebra.

The slope m tells you how steep the line is and in which direction it travels. A positive slope means the line rises from left to right; a negative slope means it falls. The absolute value of m describes exactly how many units the line rises (or falls) for every one unit you move to the right. A slope of 2, for example, means the line goes up 2 units for each 1 unit of horizontal movement — a "rise over run" ratio of 2:1.

The y-intercept b anchors the line on the coordinate plane. It is the y-value of the line when x = 0, so you can instantly plot the point (0, b) without substituting any value. From that anchor point, you apply the slope to trace the rest of the line. This makes y = mx + b the most graphing-friendly form of a linear equation, which is why it dominates secondary-school mathematics curricula worldwide and is the preferred form for cost functions, speed-distance relationships, and any other linear model.

Forms of Linear Equations Compared

Form	Equation	Best Used When
Slope-Intercept	y = mx + b	Graphing, reading slope and intercept directly
Standard Form	Ax + By = C	Integer coefficients, finding both intercepts quickly
Point-Slope	y − y₁ = m(x − x₁)	Writing an equation from one point and a known slope
Intercept Form	x/a + y/b = 1	When both x-intercept (a) and y-intercept (b) are known

How to Use This Calculator

Choose the tab matching the information you have: Standard Form, Slope-Intercept, Two Points, Point + Slope, or Parallel & Perpendicular.
Enter your values in the input fields. Decimal values are accepted for non-integer slopes and intercepts.
Click Calculate. Results appear instantly below the inputs with stat cards, both equation forms, and step-by-step working.
Scroll down to the Line Graph to see the line plotted with labeled intercepts and a rise/run slope triangle.
Use the Sample button to auto-fill an example if you are unsure what values to enter.

Conversion Formulas with Worked Examples

Standard Form to Slope-Intercept — Example

Given 2x + 3y = 12:

Subtract 2x from both sides: 3y = −2x + 12
Divide every term by B = 3: y = −(2/3)x + 4
Result: slope m = −2/3, y-intercept b = 4, x-intercept = 6

Two Points to Equation — Example

Given (1, 3) and (4, 9):

Slope: m = (9 − 3) / (4 − 1) = 6/3 = 2
Use point (1, 3): 3 = 2(1) + b ⇒ b = 1
Slope-intercept form: y = 2x + 1
Standard form: −2x + y = 1 ⇒ multiply by −1: 2x − y = −1

Point-Slope to Slope-Intercept — Example

Given point (3, 5) and slope m = 2:

Point-slope form: y − 5 = 2(x − 3)
Expand: y − 5 = 2x − 6
Add 5: y = 2x − 1. So b = −1

Graph Interpretation Guide

Positive slope (m > 0): Line rises from left to right. Steeper as m increases.
Negative slope (m < 0): Line falls from left to right. Steeper as |m| increases.
Zero slope (m = 0): Horizontal line y = b. No rise, only runs.
Undefined slope: Vertical line x = a. Cannot be written in y = mx + b form.
Y-intercept (0, b): Where the line meets the y-axis (set x = 0).
X-intercept (−b/m, 0): Where the line meets the x-axis (set y = 0 and solve for x).
Slope triangle: The orange rise/run triangle on the graph shows slope visually — a rise of m units for every 1 unit of horizontal run.

Real-World Examples of Slope-Intercept Form

Cost function: A plumber charges a $50 call-out fee plus $80 per hour. Total cost: y = 80x + 50. Slope = 80 (cost per hour), y-intercept = 50 (fixed fee).
Distance-time: A car travelling at 60 km/h starting 20 km from home: d = 60t + 20. Slope = 60 (speed in km/h), y-intercept = 20 (starting distance).
Temperature conversion: Celsius from Fahrenheit: C = (5/9)F − 160/9. Slope = 5/9, b ≈ −17.78 (the value of 0°F in Celsius).
Depreciation: A machine worth $10,000 depreciating $1,000 per year: V = −1000t + 10000. Slope = −1000, y-intercept = 10,000 (initial value).

Frequently Asked Questions

What is slope-intercept form?

Slope-intercept form is y = mx + b, where m is the slope (rise over run) and b is the y-intercept (the y-value when x = 0). It is the most common linear equation format because you can read slope and intercept directly without any additional algebra, making it ideal for graphing and analysing linear relationships in science, engineering, and everyday problem-solving.

What does m represent in y = mx + b?

m is the slope of the line — the ratio of vertical change (rise) to horizontal change (run) between any two points on the line. A positive m means the line goes up from left to right; a negative m means it descends. When m = 0 the line is horizontal. A larger absolute value of m produces a steeper line.

How do I find the slope from two points?

Use the slope formula m = (y₂ − y₁) / (x₂ − x₁). For (1, 3) and (4, 9): m = (9 − 3) / (4 − 1) = 6 / 3 = 2. Then substitute back into y = mx + b with either point to find b. Using (1, 3): 3 = 2(1) + b, so b = 1, giving y = 2x + 1.

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

Slope-intercept (y = mx + b) directly shows slope m and y-intercept b, making it ideal for graphing. Standard form (Ax + By = C) uses integer coefficients with A positive and is preferred for finding integer solutions, writing clean equations, and certain algebraic manipulations. Both forms represent the same straight line — they are fully algebraically equivalent.

What is an undefined slope (vertical line)?

A vertical line (e.g., x = 3) has an undefined slope because the horizontal change (run) between any two points is zero, making rise/run a division by zero. Vertical lines cannot be expressed in slope-intercept form y = mx + b at all. In standard form they appear simply as x = constant (e.g., 1x + 0y = 3).

What is a zero slope (horizontal line)?

A zero slope (m = 0) means the line is perfectly horizontal. The equation simplifies to y = b, where b is the constant y-value of every point on the line. There is no rise as x increases. For example, y = 5 is a horizontal line 5 units above the x-axis. Horizontal lines have no x-intercept unless b = 0.

How are parallel and perpendicular slopes related?

Parallel lines have identical slopes (m₁ = m₂) but different y-intercepts. Perpendicular lines have slopes that are negative reciprocals of each other: m₁ × m₂ = −1. For example, if a line has slope 3, a perpendicular line has slope −1/3. If a line is horizontal (m = 0), its perpendicular is a vertical line with undefined slope.

How do I convert standard form Ax + By = C to slope-intercept form?

Subtract Ax from both sides: By = −Ax + C. Then divide each term by B: y = (−A/B)x + (C/B). So slope m = −A/B and y-intercept b = C/B. Example: 2x + 3y = 12 ⇒ 3y = −2x + 12 ⇒ y = (−2/3)x + 4. If B = 0, the line is vertical and has no slope-intercept form.