# 2.2 Linear equations in one variable  (Page 4/15)

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Solve $\text{\hspace{0.17em}}\frac{-3}{2x+1}=\frac{4}{3x+1}.\text{\hspace{0.17em}}$ State the excluded values.

$x=-\frac{7}{17}.\text{\hspace{0.17em}}$ Excluded values are $\text{\hspace{0.17em}}x=-\frac{1}{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=-\frac{1}{3}.$

## Solving a rational equation with factored denominators and stating excluded values

Solve the rational equation after factoring the denominators: $\text{\hspace{0.17em}}\frac{2}{x+1}-\frac{1}{x-1}=\frac{2x}{{x}^{2}-1}.\text{\hspace{0.17em}}$ State the excluded values.

We must factor the denominator $\text{\hspace{0.17em}}{x}^{2}-1.\text{\hspace{0.17em}}$ We recognize this as the difference of squares, and factor it as $\text{\hspace{0.17em}}\left(x-1\right)\left(x+1\right).\text{\hspace{0.17em}}$ Thus, the LCD that contains each denominator is $\text{\hspace{0.17em}}\left(x-1\right)\left(x+1\right).\text{\hspace{0.17em}}$ Multiply the whole equation by the LCD, cancel out the denominators, and solve the remaining equation.

The solution is $\text{\hspace{0.17em}}x=-3.\text{\hspace{0.17em}}$ The excluded values are $\text{\hspace{0.17em}}x=1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=-1.$

Solve the rational equation: $\text{\hspace{0.17em}}\frac{2}{x-2}+\frac{1}{x+1}=\frac{1}{{x}^{2}-x-2}.$

$x=\frac{1}{3}$

## Finding a linear equation

Perhaps the most familiar form of a linear equation is the slope-intercept form, written as $\text{\hspace{0.17em}}y=mx+b,$ where $\text{\hspace{0.17em}}m=\text{slope}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b=y\text{−intercept}\text{.}\text{\hspace{0.17em}}$ Let us begin with the slope.

## The slope of a line

The slope    of a line refers to the ratio of the vertical change in y over the horizontal change in x between any two points on a line. It indicates the direction in which a line slants as well as its steepness. Slope is sometimes described as rise over run.

$m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$

If the slope is positive, the line slants to the right. If the slope is negative, the line slants to the left. As the slope increases, the line becomes steeper. Some examples are shown in [link] . The lines indicate the following slopes: $\text{\hspace{0.17em}}m=-3,$ $m=2,$ and $\text{\hspace{0.17em}}m=\frac{1}{3}.$

## The slope of a line

The slope of a line, m , represents the change in y over the change in x. Given two points, $\text{\hspace{0.17em}}\left({x}_{1},{y}_{1}\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left({x}_{2},{y}_{2}\right),$ the following formula determines the slope of a line containing these points:

$m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$

## Finding the slope of a line given two points

Find the slope of a line that passes through the points $\text{\hspace{0.17em}}\left(2,-1\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(-5,3\right).$

We substitute the y- values and the x- values into the formula.

$\begin{array}{ccc}\hfill m& =& \frac{3-\left(-1\right)}{-5-2}\hfill \\ & =& \frac{4}{-7}\hfill \\ & =& -\frac{4}{7}\hfill \end{array}$

The slope is $\text{\hspace{0.17em}}-\frac{4}{7}.$

Find the slope of the line that passes through the points $\text{\hspace{0.17em}}\left(-2,6\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(1,4\right).$

$m=-\frac{2}{3}$

## Identifying the slope and y- Intercept of a line given an equation

Identify the slope and y- intercept, given the equation $\text{\hspace{0.17em}}y=-\frac{3}{4}x-4.$

As the line is in $\text{\hspace{0.17em}}y=mx+b\text{\hspace{0.17em}}$ form, the given line has a slope of $\text{\hspace{0.17em}}m=-\frac{3}{4}.\text{\hspace{0.17em}}$ The y- intercept is $\text{\hspace{0.17em}}b=-4.$

## The point-slope formula

Given the slope and one point on a line, we can find the equation of the line using the point-slope formula.

$y-{y}_{1}=m\left(x-{x}_{1}\right)$

This is an important formula, as it will be used in other areas of college algebra and often in calculus to find the equation of a tangent line. We need only one point and the slope of the line to use the formula. After substituting the slope and the coordinates of one point into the formula, we simplify it and write it in slope-intercept form.

## The point-slope formula

Given one point and the slope, the point-slope formula will lead to the equation of a line:

$y-{y}_{1}=m\left(x-{x}_{1}\right)$

## Finding the equation of a line given the slope and one point

Write the equation of the line with slope $\text{\hspace{0.17em}}m=-3\text{\hspace{0.17em}}$ and passing through the point $\text{\hspace{0.17em}}\left(4,8\right).\text{\hspace{0.17em}}$ Write the final equation in slope-intercept form.

Using the point-slope formula, substitute $\text{\hspace{0.17em}}-3\text{\hspace{0.17em}}$ for m and the point $\text{\hspace{0.17em}}\left(4,8\right)\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}\left({x}_{1},{y}_{1}\right).$

$\begin{array}{ccc}\hfill y-{y}_{1}& =& m\left(x-{x}_{1}\right)\hfill \\ \hfill y-8& =& -3\left(x-4\right)\hfill \\ \hfill y-8& =& -3x+12\hfill \\ \hfill y& =& -3x+20\hfill \end{array}$

Need help solving this problem (2/7)^-2
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
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salma
Commplementary angles
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Sherica
im all ears I need to learn
Sherica
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Tamia
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Uday
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salma
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Ayuba
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opoku
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Ali
greetings from Iran
Ali
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Bach
hi
Nharnhar
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_