<< Chapter < Page Chapter >> Page >

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 .

Got questions? Get instant answers now!

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 −3. The excluded values are 1 and −1.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

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

x = 1 3

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

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

m = 2 3

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

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
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply
madras university algebra questions papers first year B. SC. maths
Kanniyappan Reply
Hey
Rightspect
hi
chesky
Give me algebra questions
Rightspect
how to send you
Vandna
What does this mean
Michael Reply
cos(x+iy)=cos alpha+isinalpha prove that: sin⁴x=sin²alpha
rajan Reply
cos(x+iy)=cos aplha+i sinalpha prove that: sinh⁴y=sin²alpha
rajan
cos(x+iy)=cos aplha+i sinalpha prove that: sinh⁴y=sin²alpha
rajan
is there any case that you can have a polynomials with a degree of four?
victor
***sscc.edu/home/jdavidso/math/catalog/polynomials/fourth/fourth.html
Oliver
can you solve it step b step
Ching Reply
give me some important question in tregnamentry
Anshuman
what is linear equation with one unknown 2x+5=3
Joan Reply
-4
Joel
x=-4
Joel
x=-1
Joan
I was wrong. I didn't move all constants to the right of the equation.
Joel
x=-1
Cristian
Adityasuman x= - 1
Aditya
y=x+1
gary
what is the VA Ha D R X int Y int of f(x) =x²+4x+4/x+2 f(x) =x³-1/x-1
Shadow Reply
can I get help with this?
Wayne
Are they two separate problems or are the two functions a system?
Wilson
Also, is the first x squared in "x+4x+4"
Wilson
x^2+4x+4?
Wilson
thank you
Wilson
Please see ***imgur.com/a/lpTpDZk for solutions
Wilson
f(x)=x square-root 2 +2x+1 how to solve this value
Marjun Reply
factor or use quadratic formula
Wilson
what is algebra
Ige Reply
The product of two is 32. Find a function that represents the sum of their squares.
Paul
Practice Key Terms 7

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask