3.7Slope-Intercept Form
y = mx + b — two numbers tell the whole story of a line. m is the rate of change, b is where the line starts. Master this form and you can instantly read, write, and graph any linear relationship.
Why This Matters
Slope-intercept form y = mx + b is the most used form of a linear equation — it immediately reveals the rate of change and starting value, making it essential for data analysis and modeling.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How does y = mx + b encode both the slope and y-intercept of a line, and how do you use this form to write equations and graph lines?
Lesson Overview
Slope-intercept form is y = mx + b, where m is the slope (rate of change) and b is the y-intercept (the value of y when x = 0). This form is the most common way to write a linear equation because you can immediately read the slope and y-intercept. You can write a slope-intercept equation given: (1) the slope and y-intercept directly; (2) the slope and any point on the line — substitute to find b; (3) two points — first find the slope, then find b. You can also convert from standard form Ax + By = C by solving for y.
Anatomy of y = mx + b
y = mx + b
m (slope): how steep the line is and which direction it goes. Positive m rises left to right; negative m falls.
b (y-intercept): where the line crosses the y-axis. The starting point when x = 0.
x and y: variables representing any point (x, y) on the line.
Worked Examples
Write the equation of a line with slope = 3 and y-intercept = −2.
Identify m = 3 and b = −2.
Substitute directly into y = mx + b.
y = 3x + (−2)
y = 3x − 2
Check: when x = 0, y = 3(0) − 2 = −2 ✓ (y-intercept confirmed).
Check: when x = 1, y = 3(1) − 2 = 1. Slope from (0,−2) to (1,1): (1−(−2))/(1−0) = 3 ✓.
Example 1 — y = 3x − 2
Write the equation of a line with slope = −(1/2) that passes through (0, 4).
The point (0, 4) is on the y-axis, so b = 4.
Identify m = −1/2 and b = 4.
Substitute into y = mx + b.
y = −(1/2)x + 4
Check: when x = 0, y = 4 ✓. When x = 2, y = −(1/2)(2) + 4 = 3. Slope from (0,4) to (2,3): (3−4)/(2−0) = −1/2 ✓.
Write the equation of a line with slope = 2 that passes through (3, 7).
We know m = 2 and the point (3, 7) is on the line.
Substitute into y = mx + b to find b.
7 = 2(3) + b
7 = 6 + b
b = 7 − 6 = 1
Write the equation: y = 2x + 1.
Check: y = 2(3) + 1 = 7 ✓.
Write the equation of the line through (1, 5) and (3, 11).
Step 1: Find the slope.
m = (y₂ − y₁) / (x₂ − x₁) = (11 − 5) / (3 − 1) = 6 / 2 = 3
Step 2: Find b using one point, say (1, 5).
5 = 3(1) + b → 5 = 3 + b → b = 2
Step 3: Write the equation.
y = 3x + 2
Check with (3, 11): y = 3(3) + 2 = 11 ✓.
Example 4 — y = 3x + 2
Convert 4x − 2y = 8 to slope-intercept form.
Goal: solve for y.
4x − 2y = 8
Subtract 4x from both sides: −2y = −4x + 8
Divide every term by −2: y = 2x − 4
Identify: m = 2, b = −4.
Check: substitute x = 3 into original: 4(3) − 2y = 8 → 12 − 2y = 8 → y = 2. Check in slope-intercept: y = 2(3) − 4 = 2 ✓.
Guided Practice
Answers are in the Answer Key section.
Guided Practice Video: Slope-Intercept Form
Watch the guided practice walkthrough for slope-intercept form, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Write the equation of a line with slope = −4 and y-intercept = 7.
Hint: Substitute m = −4 and b = 7 directly into y = mx + b → y = −4x + 7.
Write the equation of a line with slope = (1/3) that passes through (0, −5).
Hint: (0, −5) is the y-intercept, so b = −5. y = (1/3)x − 5.
Write the equation of a line with slope = −2 that passes through (4, 1).
Hint: Substitute (4,1) and m = −2 into y = mx + b: 1 = −2(4) + b → b = 9. y = −2x + 9.
Write the equation of the line through (0, 3) and (2, 7).
Hint: m = (7−3)/(2−0) = 4/2 = 2. b = 3 (y-intercept given). y = 2x + 3.
Convert 6x − 3y = 12 to slope-intercept form.
Hint: Subtract 6x: −3y = −6x + 12. Divide by −3: y = 2x − 4.
Key Vocabulary
Slope-Intercept Form
The equation y = mx + b, where m is the slope and b is the y-intercept.
Example: y = 3x − 2 has slope 3 and y-intercept −2.
y-intercept
The point where a line crosses the y-axis; the value of y when x = 0. In y = mx + b, the y-intercept is b.
Example: In y = 3x − 2, the y-intercept is (0, −2).
Slope
The rate of change of a line, m = rise/run = (y₂−y₁)/(x₂−x₁). In y = mx + b, the slope is m.
Example: In y = 3x − 2, the slope is 3 (up 3, right 1).
Linear Model
A linear equation used to represent a real-world relationship. The slope is the rate of change and b is the initial value.
Example: C = 0.10t + 30 models a phone plan: $30 base + $0.10 per text.
Point-Slope Form
An alternate form y − y₁ = m(x − x₁) used when you know a point and the slope. Can be converted to slope-intercept form.
Example: Through (3,7) with m=2: y−7 = 2(x−3) → y = 2x+1.
Interactive Practice — 5 Questions
What is the slope of the line y = −5x + 3?
What is the y-intercept of y = 4x − 7?
Which equation has slope 2 and passes through (0, −3)?
Convert 3x − y = 5 to slope-intercept form.
A line passes through (2, 9) and (5, 18). What is the equation?
Independent Practice
Answers are in the Answer Key section.
Independent Practice
Write the equation of a line with slope = 5 and y-intercept = −1.
Write the equation of a line with slope = −(3/4) and y-intercept = 6.
Write the equation of a line with slope = 4 that passes through (2, 11).
Write the equation of the line through (−1, 2) and (3, 10).
Challenge: A line passes through (−2, 5) and (4, −1). Write the equation in slope-intercept form, then convert it to standard form Ax + By = C.
ChallengeCommon Mistakes
Reading b as the slope and m as the y-intercept in y = mx + b.
m is always the coefficient of x (slope). b is always the constant term (y-intercept). In y = 4x − 7, m = 4 and b = −7.
When finding b from a point, substituting into the wrong variable — e.g., using x where y should go.
Substitute the point's x-value for x and y-value for y, then solve for b. For point (3,7) with m=2: 7 = 2(3) + b.
Forgetting to flip the sign when converting from standard form — e.g., writing y = 3x + 5 instead of y = 3x − 5 for 3x − y = 5.
When you multiply or divide by a negative number, all signs flip. Solve step by step: −y = −3x + 5 → y = 3x − 5.
Writing the equation as y = b + mx instead of y = mx + b.
Both are mathematically equivalent, but the standard convention is y = mx + b with the slope term first. Use this form to avoid confusion.
Math Tips
To write an equation from a graph: read the y-intercept where the line crosses the y-axis, then count rise/run to find the slope.
When given two points, always find slope first (m = Δy/Δx), then substitute one point to find b.
A quick check: substitute both given points into your final equation. Both should satisfy it.
In real-world models, b is the starting value (initial amount) and m is the rate of change (how fast it grows or shrinks).