1. Home
  2. Algebra and trigonometry
  3. Equations and inequalities
  4. Linear equations in one variable

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

<< Chapter < Page
  Algebra and trigonometry     Page 4 / 15
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

what does mean opportunity cost?
Aster Reply
what is poetive effect of population growth
Solomon Reply
what is inflation
Nasir Reply
what is demand
Eleni
what is economics
IMLAN Reply
economics theory describes individual behavior as the result of a process of optimization under constraints the objective to be reached being determined by
Kalkidan
Economics is a branch of social science that deal with How to wise use of resource ,s
Kassie
need
WARKISA
Economic Needs: In economics, needs are goods or services that are necessary for maintaining a certain standard of living. This includes things like healthcare, education, and transportation.
Kalkidan
What is demand and supply
EMPEROR Reply
deman means?
Alex
what is supply?
Alex
ex play supply?
Alex
Money market is a branch or segment of financial market where short-term debt instruments are traded upon. The instruments in this market includes Treasury bills, Bonds, Commercial Papers, Call money among other.
murana Reply
good
Kayode
what is money market
umar Reply
Examine the distinction between theory of comparative cost Advantage and theory of factor proportion
Fatima Reply
What is inflation
Bright Reply
a general and ongoing rise in the level of prices in an economy
AI-Robot
What are the factors that affect demand for a commodity
Florence Reply
price
Kenu
differentiate between demand and supply giving examples
Lambiv Reply
differentiated between demand and supply using examples
Lambiv
what is labour ?
Lambiv
how will I do?
Venny Reply
how is the graph works?I don't fully understand
Rezat Reply
information
Eliyee
devaluation
Eliyee
t
WARKISA
hi guys good evening to all
Lambiv
multiple choice question
Aster Reply
appreciation
Eliyee
explain perfect market
Lindiwe Reply
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
Ezea
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 7

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




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