Lines, Slope and Linear Equations

Part 1 · Hook

Why this matters

A road sign that warns of a 6% grade means the road drops 6 feet for every 100 feet you drive forward. That ratio is a slope. The next topics measure how fast any function changes by drawing lines through two of its points, so slope and the equation of a line need to be quick and sure.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. Solve $3x - 4 = 11$ .

$5$
$\frac{7}{3}$
$15$
$-5$

Show the answer

$3x = 15$ , so $x = 5$ .

Correct: $5$ :
$\frac{7}{3}$ :
$15$ :
$-5$ :

2. For $f(x) = x^2$ , what is the point $(3, f(3))$ ?

$(3, 9)$
$(3, 6)$
$(9, 3)$
$(3, 3)$

Show the answer

$f(3) = 9$ , so the point is $(3, 9)$ .

Correct: $(3, 9)$ :
$(3, 6)$ :
$(9, 3)$ :
$(3, 3)$ :

Part 4 · See it

See it first

The line through (1, 2) and (4, 8). A dashed horizontal run of 3 goes from (1, 2) to (4, 2) and a dashed vertical rise of 6 goes up to (4, 8), so the slope is 6/3 = 2. — The line through $(1, 2)$ and $(4, 8)$ rises $6$ over a run of $3$ : slope $\frac{6}{3} = 2$ . LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

Two points $(x_1, y_1)$ and $(x_2, y_2)$ lie on a lineits slope is $\frac{y_2 - y_1}{x_2 - x_1}$ , rise over run, subtracted in the same order
You know one point $(x_1, y_1)$ and the slope $m$ point-slope form $y - y_1 = m(x - x_1)$ gives the line, and $y = y_1 + m(x - x_1)$ gives any value on it
$x$ increases by $\Delta x$ along a line of slope $m$ $y$ changes by $m \cdot \Delta x$ , the same for every equal step: a line has a constant rate of change

Part 6 · Key ideas

Key ideas

Slope $m$ $= \frac{\text{change in } y}{\text{change in } x}$ : positive rises, negative falls, $0$ is horizontal, a vertical line has none.
Point-slope form $y - y_1 = m(x - x_1)$ ; slope-intercept form $y = mx + b$ with $b$ the $y$ -intercept.
In context the slope has units of output per input, like dollars per minute.

Part 7 · Misconception

A common mistake

The wrong idea: In $y - 3 = 2(x - 1)$ , the $y$ -intercept is $3$ .

What actually happens: $3$ is the $y$ -value at $x = 1$ , not at $x = 0$ . Solving gives $y = 2x + 1$ , so the $y$ -intercept is $1$ .

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

1. What is the slope of the line through $(-2, 5)$ and $(4, -7)$ ?

$2$
$-\frac{1}{2}$
$-2$
$-6$

Show the answer

Slope $= \frac{y_2 - y_1}{x_2 - x_1}$ $= \frac{-7 - 5}{4 - (-2)}$ $= \frac{-12}{6}$ $= -2$ . Subtract in the same order on top and bottom.

$2$ : The $y$ -values fall from $5$ to $-7$ as $x$ rises, so the slope is negative.
$-\frac{1}{2}$ : This is run over rise. Slope is rise over run: $\frac{-12}{6}$ .
Correct: $-2$ : Right: $\frac{-7 - 5}{4 - (-2)}$ $= \frac{-12}{6}$ $= -2$ .
$-6$ : This subtracts the $y$ -values, $-7 - 5 = -12$ , but adds the $x$ -values, $4 + (-2) = 2$ . Subtract both in the same order: $4 - (-2) = 6$ .

2. Write an equation of the line through $(1, 3)$ with slope $2$ . Solve for $y$ .

Type an expression for $y =$ .

Show the answer

Point-slope form: $y - 3 = 2(x - 1)$ , so $y$ $= 3 + 2(x - 1)$ $= 2x + 1$ . Check: at $x = 1$ , $y = 3$ .

Answer: $y =$ $3 + 2(x - 1)$

3. A phone plan costs $C(m) = 15 + 0.10m$ dollars for $m$ minutes of calls. What does the slope $0.10$ mean?

Each extra minute adds $0.10 to the cost.
The plan costs $0.10 with no calls.
Each extra dollar buys $0.10$ minute.
The cost rises $15 for each minute.

Show the answer

In $C(m) = 15 + 0.10m$ , the slope $0.10$ is the change in $C$ per $1$ -unit change in $m$ : $0.10 per minute. The $15 is the cost at $m = 0$ .

Correct: Each extra minute adds $0.10 to the cost.: Right: the slope is the change in cost for each $1$ -minute increase in $m$ .
The plan costs $0.10 with no calls.: With no calls ( $m = 0$ ) the cost is $15, the intercept.
Each extra dollar buys $0.10$ minute.: This swaps input and output; the slope is dollars per minute.
The cost rises $15 for each minute.: $15 is the starting cost, not the change per minute.

4. A line passes through $(2, 10)$ and $(6, 22)$ . What is $y$ on this line at $x = 3$ ?

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

Slope $= \frac{22 - 10}{6 - 2} = 3$ . From $(2, 10)$ : $y = 10 + 3(x - 2)$ , so at $x = 3$ , $y = 13$ .

Answer: 13

5. For $f(x) = x^2$ , what is the slope of the line through $(2, f(2))$ and $(5, f(5))$ ?

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

The points are $(2, 4)$ and $(5, 25)$ . Slope $= \frac{25 - 4}{5 - 2}$ $= \frac{21}{3}$ $= 7$ .

Answer: 7

6. Which equation is the same line as $y - 4 = -3(x + 2)$ ?

$y = -3x - 2$
$y = -3x + 10$
$y = -3x - 10$
$y = -3x + 2$

Show the answer

Distribute: $y - 4 = -3x - 6$ . Add $4$ : $y = -3x - 2$ . Check: at $x = -2$ , $y = 6 - 2 = 4$ .

Correct: $y = -3x - 2$ : Right: $y$ $= -3x - 6 + 4$ $= -3x - 2$ .
$y = -3x + 10$ : Distribute the $-3$ to both terms: $-3(x + 2) = -3x - 6$ , not $-3x + 6$ .
$y = -3x - 10$ : After distributing, $4$ is added to both sides: $-6 + 4 = -2$ .
$y = -3x + 2$ : Check the point $(-2, 4)$ : $-3(-2) + 2 = 8$ , not $4$ .

Part 9 · Summary

Summary

Slope is the change in

y

divided by the change in

x

, the same between any two points of a line. Point-slope form

y - y_1 = m(x - x_1)

builds a line from one point and the slope, and

y = y_1 + m(x - x_1)

gives its value at any

x

. In context, slope is output units per input unit.

Part 10 · Up next

What comes next

Part 11 · Connections