Find the equation of the line parallel to
$\text{\hspace{0.17em}}5x=7+y\text{\hspace{0.17em}}$ and passing through the point
$\text{\hspace{0.17em}}\left(\mathrm{-1},\mathrm{-2}\right).$
We see that the slope is
$\text{\hspace{0.17em}}m=\frac{5}{3}.\text{\hspace{0.17em}}$ This means that the slope of the line perpendicular to the given line is the negative reciprocal, or
$-\frac{3}{5}.\text{\hspace{0.17em}}$ Next, we use the point-slope formula with this new slope and the given point.
We can solve linear equations in one variable in the form
$\text{\hspace{0.17em}}ax+b=0\text{\hspace{0.17em}}$ using standard algebraic properties. See
[link] and
[link].
A rational expression is a quotient of two polynomials. We use the LCD to clear the fractions from an equation. See
[link] and
[link].
All solutions to a rational equation should be verified within the original equation to avoid an undefined term, or zero in the denominator. See
[link] and
[link].
Given two points, we can find the slope of a line using the slope formula. See
[link] .
We can identify the slope and
y -intercept of an equation in slope-intercept form. See
[link].
We can find the equation of a line given the slope and a point. See
[link].
We can also find the equation of a line given two points. Find the slope and use the point-slope formula. See
[link].
The standard form of a line has no fractions. See
[link] .
Horizontal lines have a slope of zero and are defined as
$\text{\hspace{0.17em}}y=c,$ where
c is a constant.
Vertical lines have an undefined slope (zero in the denominator), and are defined as
$\text{\hspace{0.17em}}x=c,$ where
c is a constant. See
[link].
Parallel lines have the same slope and different
y- intercepts. See
[link] .
Perpendicular lines have slopes that are negative reciprocals of each other unless one is horizontal and the other is vertical. See
[link] .
Section exercises
Verbal
What does it mean when we say that two lines are parallel?
explain why we must exclude
$\text{\hspace{0.17em}}x=5\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}x=\mathrm{-1}\text{\hspace{0.17em}}$ as possible solutions from the solution set.
If we insert either value into the equation, they make an expression in the equation undefined (zero in the denominator).
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5) and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.