Before you read

Answer first — the lesson below explains it.

A business model is expressed as the equation $y = - 5 x + 12$ , where $x$ is months and $y$ is revenue. What are the slope (monthly change) and y-intercept (initial revenue)?

A
Slope $= 12$ , y-intercept $= - 5$
B
Slope $= - 5$ , y-intercept $= 12$
C
Slope $= 5$ , y-intercept $= 12$
D
Slope $= - 5$ , y-intercept $= - 12$

Check your answer

Answer:

Before you read

Answer first — the lesson below explains it.

A cost equation $3 x + y = 9$ models expenses. What is the slope (rate of cost change)?

A
$3$
B
$- 3$
C
$9$
D
$\frac{1}{3}$

Check your answer

Answer:

Theory

The Slope-Intercept Form

The most common way to write a linear equation is slope-intercept form:

$y = mx + b$

where:
- $m$ = slope (rate of change)
- $b$ = y-intercept (where the line crosses the y-axis)

y = ½x + 2: slope m = ½, y-intercept b = 2

This form is powerful because you can read the slope and y-intercept directly from the equation without any calculation.

Example 1

Identify the slope and y-intercept of $y = 3x - 4$ .

Comparing with $y = mx + b$ :

Slope: $m = 3$

Y-intercept: $b = -4$ (the point $(0, -4)$ )

m = 3

b = -4

Example 2

Identify the slope and y-intercept of $y = -\frac{2}{3}x + 5$ .

Slope: $m = -\frac{2}{3}$

Y-intercept: $b = 5$ (the point $(0, 5)$ )

m = -\frac{2}{3}

b = 5

Quick check

You just read it. Can you apply it?

A profit trend has slope $- 2$ and y-intercept $7$ . What is the equation in slope-intercept form?

A
$y = 7 x - 2$
B
$y = - 2 x + 7$
C
$y = 2 x + 7$
D
$y = - 2 x - 7$

Check your answer

Answer:

Plug directly into $y = m x + b$ : $y = - 2 x + 7$ .

Theory

Writing an Equation from Slope and a Point

If you know the slope and one point on the line, you can write the equation:

1. Start with $y = mx + b$
2. Plug in $m$ (slope) and the point $(x, y)$
3. Solve for $b$
4. Write the final equation

Example 1

Write the equation of a line with slope $4$ that passes through $(2, 11)$ .

Start: $y = 4x + b$

Plug in $(2, 11)$ : $11 = 4(2) + b$

$11 = 8 + b$

$b = 3$

Equation: $y = 4x + 3$

y = 4x + 3

Example 2

Write the equation of a line through $(1, 2)$ and $(3, 8)$ .

Step 1: Find slope: $m = \frac{8 - 2}{3 - 1} = \frac{6}{2} = 3$

Step 2: Use $y = 3x + b$ with point $(1, 2)$ :

$2 = 3(1) + b$ $\to$ $b = -1$

Equation: $y = 3x - 1$

y = 3x - 1

Quick check

You just read it. Can you apply it?

A line has slope $3$ and passes through $(1, 5)$ . What is its equation in slope-intercept form?

A
$y = 3 x + 2$
B
$y = 3 x + 5$
C
$y = 3 x - 2$
D
$y = 5 x + 3$

Check your answer

Answer:

$y = 3 x + b$ . Plug in $(1, 5)$ : $5 = 3 (1) + b$ $\to$ $b = 2$ .
Equation: $y = 3 x + 2$ .

Theory

Graphing from Slope-Intercept Form

To graph $y = mx + b$ :
1. Plot the y-intercept $(0, b)$
2. Use the slope to find a second point: from $(0, b)$ , go right by the run and up/down by the rise
3. Draw the line through the two points

Example: For $y = \frac{2}{3}x - 1$ :
1. Plot $(0, -1)$
2. From there, go right 3, up 2 $\to$ plot $(3, 1)$
3. Draw the line

Quick check

You just read it. Can you apply it?

Which equation represents the line through $(- 2, 1)$ and $(4, 7)$ ?

A
$y = x + 3$
B
$y = x - 3$
C
$y = 2 x + 5$
D
$y = \frac{1}{2} x + 2$

Check your answer

Answer:

Slope: $m = \frac{7 - 1}{4 - ( - 2 )} = \frac{6}{6} = 1$
Using $(4, 7)$ : $7 = 1 (4) + b$ $\to$ $b = 3$ .
Equation: $y = x + 3$ .

Common Mistakes

Common Mistakes to Avoid

Reading the wrong sign for

b

In $y = 2x - 7$ , thinking $b = 7$ instead of $b = -7$

The equation is

y = 2x + (-7)

. The y-intercept is

-7

Confusing

y = 5x

with 'no y-intercept'

Thinking $y = 5x$ has no y-intercept

y = 5x

is really

y = 5x + 0

. The y-intercept is

0

(the line passes through the origin).

Forgetting to solve for

y

first

Reading slope from $2x + y = 6$ as $m = 2$

Rearrange first:

y = -2x + 6

, so

m = -2

Quick check

You just read it. Can you apply it?

What is the slope of the line $2 x - 4 y = 12$ ?

A
$2$
B
$- 4$
C
$\frac{1}{2}$
D
$- 3$

Check your answer

Answer:

Solve for $y$ :
$- 4 y = - 2 x + 12$
$y = \frac{- 2 x + 12}{- 4} = \frac{1}{2} x - 3$
Slope: $m = \frac{1}{2}$ .

Strategy

SAT Strategy Tip

Many SAT questions give an equation in non-standard form and ask for slope or y-intercept. Always rearrange to $y = mx + b$ first — it takes 10 seconds and makes the answer obvious. Also, when matching a graph to an equation, check the y-intercept first (it's the quickest check) then verify the slope.

After reading

Same question as before. What do you say now?

A business model is expressed as the equation $y = - 5 x + 12$ , where $x$ is months and $y$ is revenue. What are the slope (monthly change) and y-intercept (initial revenue)?

A
Slope $= 12$ , y-intercept $= - 5$
B
Slope $= - 5$ , y-intercept $= 12$
C
Slope $= 5$ , y-intercept $= 12$
D
Slope $= - 5$ , y-intercept $= - 12$

Check your answer

Answer:

In $y = m x + b$ : $m = - 5$ (slope), $b = 12$ (y-intercept).

After reading

Same question as before. What do you say now?

A cost equation $3 x + y = 9$ models expenses. What is the slope (rate of cost change)?

A
$3$
B
$- 3$
C
$9$
D
$\frac{1}{3}$

Check your answer

Answer:

Rearrange: $y = - 3 x + 9$ . The slope is $m = - 3$ .

Ready to ace the SAT?

Try NovaMath free — AI tutoring, 115 lessons, 2,400+ exercises.

Try NovaMath Free