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).