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Slope & Intercept
\( y = mx + b \)

Learn how slope measures rate of change and the y-intercept reveals the starting value: the two keys to reading any linear equation.

Pre-Algebra

What is a Linear Equation?

A linear equation describes a straight-line relationship between two quantities. The most common form is called slope-intercept form:

\( m \)
The slope: how steep the line is and how fast \( y \) changes
\( b \)
The y-intercept: where the line crosses the y-axis

Visualizing \( y = 2x + 1 \)

x y 0 1 2 3 4 5 6 0 1 2 3 4 5 6
\((0, 1)\)
\((1, 3)\)
\((2, 5)\)
 +2 +1
\(b = 1\)

Slope as Rate of Change

The slope tells you how much the output changes per unit of input. Think of it as a speed or rate.

Positive slope
Line goes up from left to right. Values increase.
Negative slope
Line goes down from left to right. Values decrease.
Zero slope
A perfectly flat horizontal line. No change at all.

Intercept as Initial Value

The y-intercept \( b \) answers the question: "What is the value when \( x = 0 \)?" In real life, it is often a starting amount before any change occurs.

x y
\( b = 3 \)
 0 1 2 3 4 5 0 3 When x = 0, y starts at 3

The Slope-Intercept Form

1
\( m \) (slope): the constant rate of change: how many units \( y \) changes for each 1-unit increase in \( x \).
2
\( b \) (y-intercept): the initial value of \( y \) when \( x = 0 \). It is where the line touches the y-axis.
3
\( x \) is the independent variable (input), and \( y \) is the dependent variable (output).

Positive vs Negative Slope

Positive: \( m > 0 \)
 Line rises left to right
Negative: \( m < 0 \)
 Line falls left to right

Slope vs Intercept: What's the Difference?

Slope \( m \) Intercept \( b \)
Meaning Rate of change Starting value
Question it answers "How fast?" "Where does it start?"
On the graph Tilt / steepness Point on y-axis
Units output per input same unit as \( y \)
Example \( \$5 \)/hour \( \$20 \) starting fee

How to Find Slope & Intercept

1
Identify two points on the line, such as \( (x_1, y_1) \) and \( (x_2, y_2) \).
2
Calculate the slope: \( m = \dfrac{y_2 - y_1}{x_2 - x_1} \). Subtract the \( y \)-values on top, \( x \)-values on the bottom.
3
Find the y-intercept: look at the table or graph for the \( y \)-value when \( x = 0 \). That's \( b \).
4
Write the equation: plug \( m \) and \( b \) into \( y = mx + b \).

Example 1: Reading a Graph

A line passes through \( (0, 4) \) and \( (3, 10) \). Find the slope and write the equation.

x y 0 1 2 3 4 5 0 4 6 8 10
\((0, 4)\)
\((3, 10)\)
 +6 +3

Example 2: From a Table

A phone plan's monthly cost depends on the number of gigabytes used:

GB used (\( x \)) Cost in $ (\( y \))
015
225
540
855

Example 3: Negative Slope in Context

A candle is 18 cm tall and burns down 2 cm per hour. Write an equation for height \( h \) after \( t \) hours.

t (hours) h (cm) 0 1 2 3 4 5 6 7 8 9 0 18
\((0, 18)\)
\((9, 0)\)

Interpreting \( m \) and \( b \) in Context

When you read a word problem, connect the math to the real-world meaning:

1
Identify the variables: What does \( x \) represent? What does \( y \) represent?
2
State the slope in words: "For every 1 [\( x \)-unit], the [\( y \)-quantity] changes by \( m \) [\( y \)-units]."
3
State the intercept in words: "When [\( x \)-quantity] is zero, the [\( y \)-quantity] is \( b \) [\( y \)-units]."

Common Mistakes to Avoid

✕ Swapping Rise & Run
Putting the \( x \)-difference on top instead of the \( y \)-difference. Remember: rise (\( y \)) over run (\( x \)).
✕ Forgetting the Sign
If the line goes down, the slope is negative. Always check direction before writing \( m \).
✕ Misreading \( b \)
The y-intercept is where \( x = 0 \), not where \( y = 0 \). Don't confuse the y-intercept with the x-intercept.

Summary: Slope & Intercept

Concept Symbol Formula / Location Real-World Meaning
Slope \( m \) \( \dfrac{y_2 - y_1}{x_2 - x_1} \) Rate of change per unit
Y-Intercept \( b \) Value of \( y \) when \( x = 0 \) Starting / initial value
Equation \( y = mx + b \) Slope-intercept form Predicts output from input

Question 1 of 5

A gym membership costs $25 per month plus a $50 sign-up fee. Which equation models the total cost \( C \) after \( m \) months?

Question 2 of 5

A line passes through \( (2, 7) \) and \( (6, 19) \). What is the slope?

Question 3 of 5

In the equation \( y = -3x + 12 \), what does the \( 12 \) represent?

Question 4 of 5

A pool has 1200 gallons of water and drains at 80 gallons per hour. The equation is \( W = -80t + 1200 \). What does the slope \( -80 \) mean?

Question 5 of 5

Which table represents a line with a slope of \( 4 \) and a y-intercept of \( 3 \)?