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Solve 3 2 x + 1 = 4 3 x + 1 . State the excluded values.

x = 7 17 . Excluded values are x = 1 2 and x = 1 3 .

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Solving a rational equation with factored denominators and stating excluded values

Solve the rational equation after factoring the denominators: 2 x + 1 1 x 1 = 2 x x 2 1 . State the excluded values.

We must factor the denominator x 2 −1. We recognize this as the difference of squares, and factor it as ( x 1 ) ( x + 1 ) . Thus, the LCD that contains each denominator is ( x 1 ) ( x + 1 ) . Multiply the whole equation by the LCD, cancel out the denominators, and solve the remaining equation.

( x 1 ) ( x + 1 ) [ 2 x + 1 1 x 1 ] = [ 2 x ( x 1 ) ( x + 1 ) ] ( x 1 ) ( x + 1 ) 2 ( x 1 ) 1 ( x + 1 ) = 2 x 2 x 2 x 1 = 2 x Distribute the negative sign . −3 x = 0 −3 = x

The solution is x = −3. The excluded values are x = 1 and x = −1.

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Solve the rational equation: 2 x 2 + 1 x + 1 = 1 x 2 x 2 .

x = 1 3

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Finding a linear equation

Perhaps the most familiar form of a linear equation is the slope-intercept form, written as y = m x + b , where m = slope and b = y −intercept . 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 = 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: m = −3 , m = 2 , and m = 1 3 .

Coordinate plane with the x and y axes ranging from negative 10 to 10.  Three linear functions are plotted: y = negative 3 times x minus 2; y = 2 times x plus 1; and y = x over 3 plus 2.

The slope of a line

The slope of a line, m , represents the change in y over the change in x. Given two points, ( x 1 , y 1 ) and ( x 2 , y 2 ) , the following formula determines the slope of a line containing these points:

m = 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 ( 2 , −1 ) and ( −5 , 3 ) .

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

m = 3 ( −1 ) −5 2 = 4 −7 = 4 7

The slope is 4 7 .

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Find the slope of the line that passes through the points ( −2 , 6 ) and ( 1 , 4 ) .

m = 2 3

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

Identify the slope and y- intercept, given the equation y = 3 4 x 4.

As the line is in y = m x + b form, the given line has a slope of m = 3 4 . The y- intercept is b = −4.

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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 ( x x 1 )

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 ( x x 1 )

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

Write the equation of the line with slope m = −3 and passing through the point ( 4 , 8 ) . Write the final equation in slope-intercept form.

Using the point-slope formula, substitute −3 for m and the point ( 4 , 8 ) for ( x 1 , y 1 ) .

y y 1 = m ( x x 1 ) y 8 = −3 ( x 4 ) y 8 = −3 x + 12 y = −3 x + 20
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Questions & Answers

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?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
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
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
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
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
Practice Key Terms 7

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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