# 2.2 Linear equations in one variable  (Page 7/15)

 Page 7 / 15

Find the equation of the line parallel to $\text{\hspace{0.17em}}5x=7+y\text{\hspace{0.17em}}$ and passing through the point $\text{\hspace{0.17em}}\left(-1,-2\right).$

$y=5x+3$

## Finding the equation of a line perpendicular to a given line passing through a given point

Find the equation of the line perpendicular to $\text{\hspace{0.17em}}5x-3y+4=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(-4,1\right).$

The first step is to write the equation in slope-intercept form.

$\begin{array}{ccc}\hfill 5x-3y+4& =& 0\hfill \\ \hfill -3y& =& -5x-4\hfill \\ \hfill y& =& \frac{5}{3}x+\frac{4}{3}\hfill \end{array}$

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.

$\begin{array}{ccc}\hfill y-1& =& -\frac{3}{5}\left(x-\left(-4\right)\right)\hfill \\ \hfill y-1& =& -\frac{3}{5}x-\frac{12}{5}\hfill \\ \hfill y& =& -\frac{3}{5}x-\frac{12}{5}+\frac{5}{5}\hfill \\ \hfill y& =& -\frac{3}{5}x-\frac{7}{5}\hfill \end{array}$

Access these online resources for additional instruction and practice with linear equations.

## Key concepts

• 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] .

## Verbal

What does it mean when we say that two lines are parallel?

It means they have the same slope.

What is the relationship between the slopes of perpendicular lines (assuming neither is horizontal nor vertical)?

How do we recognize when an equation, for example $\text{\hspace{0.17em}}y=4x+3,$ will be a straight line (linear) when graphed?

The exponent of the $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ variable is 1. It is called a first-degree equation.

What does it mean when we say that a linear equation is inconsistent?

When solving the following equation:

$\frac{2}{x-5}=\frac{4}{x+1}$

explain why we must exclude $\text{\hspace{0.17em}}x=5\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=-1\text{\hspace{0.17em}}$ 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).

## Algebraic

For the following exercises, solve the equation for $\text{\hspace{0.17em}}x.$

$7x+2=3x-9$

$4x-3=5$

$x=2$

$3\left(x+2\right)-12=5\left(x+1\right)$

$12-5\left(x+3\right)=2x-5$

$x=\frac{2}{7}$

$\frac{1}{2}-\frac{1}{3}x=\frac{4}{3}$

$\frac{x}{3}-\frac{3}{4}=\frac{2x+3}{12}$

$x=6$

$\frac{2}{3}x+\frac{1}{2}=\frac{31}{6}$

$3\left(2x-1\right)+x=5x+3$

$x=3$

$\frac{2x}{3}-\frac{3}{4}=\frac{x}{6}+\frac{21}{4}$

given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
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nothing up todat yet
Miranda
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jai
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jai
Miranda Drice
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jai
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Miranda
I am living in india
jai
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Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
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Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
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jai
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jai
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Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
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Jeffrey
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Miranda
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Jeffrey
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Miranda
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Jeffrey
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Miranda
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Miranda
Jeffrey
Jeffrey
Miranda
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Miranda
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Steve
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Steve
I don't know why. But Im trying to like it.
Jeffrey
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Jeffrey
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Miranda
what is the solution of the given equation?
which equation
Miranda
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Jeffrey
Miranda
Jeffrey
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
cos(- z)=cos z
Mustafa
what is a algebra
(x+x)3=?
6x
Obed
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
Hi
Abdel
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Ye
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Nokwanda
C'est comment
Abdel
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Amanda
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SORIE
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Ranjay
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ANSHU
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Chinni
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Hassan
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SORIE
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SORIE
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Yaona
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SORIE
it's 12
what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls