1.10 Lines

1.10.1 Learning Objectives

By the end of this section, you should be able to:

calculate and interpret slope;
write equations of lines in several forms;
graph linear equations;
work with horizontal and vertical lines; and
find equations of parallel and perpendicular lines.

1.10.2 Slope

The slope of a nonvertical line measures its rate of change:

\[ m=\frac{\text{rise}}{\text{run}} =\frac{y_2-y_1}{x_2-x_1}. \]

A positive slope rises from left to right, a negative slope falls, and a zero slope is horizontal. A vertical line has undefined slope.

Example 1.48 Slope through Two Points

For $A(-1,4)$ and $B(5,-2)$,

\[ m=\frac{-2-4}{5-(-1)}=\boxed{-1}. \]

Exercise 1.54 Find the slope through $(-2,1)$ and $(4,10)$.

1.10.3 Equations of Lines

Common equation forms are:

\[ \text{Point-slope:}\qquad y-y_1=m(x-x_1), \]

Example 1.49 Point and Slope

The line through $(3,-1)$ with slope $2$ is

\[ y+1=2(x-3), \]

\[ \boxed{y=2x-7}. \]

Exercise 1.55 Write the line through $(1,5)$ with slope $-3$.

\[ \text{Slope-intercept:}\qquad y=mx+b, \]

\[ \text{General form:}\qquad Ax+By+C=0. \]

Example 1.50 Two Points

The line through $(2,5)$ and $(6,13)$ has slope $2$. Therefore,

\[ y-5=2(x-2), \]

so $\boxed{y=2x+1}$.

Exercise 1.56 Find the slope and intercept of $2x+5y=15$.

Exercise 1.57 Write the horizontal line through $(7,-4)$.

1.10.4 Horizontal, Vertical, Parallel, and Perpendicular Lines

Horizontal lines have equations $y=b$. Vertical lines have equations $x=a$.

Two distinct nonvertical lines are parallel when their slopes are equal. They are perpendicular when their slopes are negative reciprocals:

\[ m_1m_2=-1. \]

Example 1.51 Parallel Line

Find the line through $(4,1)$ parallel to $3x-2y=8$.

The given line has slope $3/2$, so

\[ y-1=\frac32(x-4). \]

Thus $\boxed{y=\frac32x-5}$.

Example 1.52 Perpendicular Line

A line perpendicular to $y=-\frac14x+6$ has slope $4$. Through $(2,3)$, its equation is

\[ y-3=4(x-2), \]

or $\boxed{y=4x-5}$.

Exercise 1.58 Find the slope of a line perpendicular to slope $2/3$.

1.10.5 Linear Models and Rates of Change

In a linear model, slope is the change in the dependent variable for each one-unit increase in the independent variable. The intercept is the predicted dependent value when the independent variable is zero.

Example 1.53 Slope as Rate of Change

A storage tank contains $V=120+18t$ litres after $t$ minutes. The slope $18$ means the volume increases by $18$ L/min, while $120$ is the initial volume.

1.10.6 Conceptual Takeaways

Slope measures vertical change per unit of horizontal change.
Different equation forms emphasize different information.
Two points determine a unique line.
Parallel lines share a slope.
Perpendicular slopes are negative reciprocals.
In applications, slope carries units and describes a rate.

1.10.7 Skills You Should Be Able to Do

Calculate and interpret slope.
Write a line from points, slope, or intercept information.
Convert among point-slope, slope-intercept, and general forms.
Graph vertical, horizontal, and oblique lines.
Find equations of parallel and perpendicular lines.
Interpret slopes and intercepts in models.

1.10.8 Practice Problems with Solutions

Find the slope through $(-4,7)$ and $(2,-5)$.

Show Solution

\[ m=\frac{-5-7}{2-(-4)}=\boxed{-2}. \]
Write the equation through $(-2,3)$ with slope $5/2$.

Show Solution

\[ y-3=\frac52(x+2), \] so $\boxed{y=\frac52x+8}$.
Write the equation through $(1,-2)$ and $(5,6)$.

Show Solution

\[ m=\frac{6-(-2)}{5-1}=2. \] \[ y+2=2(x-1), \] so $\boxed{y=2x-4}$.
Find the intercepts of $4x+3y=24$.

Show Solution

Setting $y=0$ gives $x=6$. Setting $x=0$ gives $y=8$. The intercepts are $\boxed{(6,0),(0,8)}$.
Write the line through $(3,4)$ parallel to $2x+y=7$.

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The given line has slope $-2$: \[ y-4=-2(x-3), \] so $\boxed{y=-2x+10}$.
Write the line through $(-1,2)$ perpendicular to $y=3x-5$.

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The perpendicular slope is $-1/3$: \[ y-2=-\frac13(x+1), \] so $\boxed{y=-\frac13x+\frac53}$.
A subscription cost is $C=9n+35$, where $n$ is the number of months. Interpret the slope and intercept.

Show Solution

The slope $9$ is a monthly cost of $\$9$. The intercept $35$ is the initial fixed fee.