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

<< Chapter < Page

College algebra

Page 4 / 15

Chapter >> Page >

Try It

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

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

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: $\frac{2}{x + 1} - \frac{1}{x - 1} = \frac{2 x}{x^{2} - 1} .$ 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.

\begin{matrix} (x - 1) (x + 1) [\frac{2}{x + 1} - \frac{1}{x - 1}] & = & [\frac{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 \end{matrix}

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

Got questions? Get instant answers now!

Try It

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

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

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 = \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: $m = −3,$ $m = 2,$ and $m = \frac{1}{3} .$

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.

A General Note

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 = \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 $(2, −1)$ and $(−5, 3) .$

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

\begin{matrix} m & = & \frac{3 - (−1)}{−5 - 2} \\ = & \frac{4}{−7} \\ = & - \frac{4}{7} \end{matrix}

The slope is $- \frac{4}{7} .$

Got questions? Get instant answers now!

Try It

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

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

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 $y = - \frac{3}{4} x - 4.$

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

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.

A General Note

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 $(x_{1}, y_{1}) .$

\begin{matrix} y - y_{1} & = & m (x - x_{1}) \\ y - 8 & = & −3 (x - 4) \\ y - 8 & = & −3 x + 12 \\ y & = & −3 x + 20 \end{matrix}

Got questions? Get instant answers now!

Questions & Answers

find the equation of the tangent to the curve y=2x³-x²+3x+1 at the points x=1 and x=3

Esther Reply

derivative of logarithms function

Iqra Reply

how to solve this question

sidra

ex 2.1 question no 11

khansa

anyone can help me

khansa

question please

Rasul

ex 2.1 question no. 11

khansa

i cant type here

khansa

Find the derivative of g(x)=−3.

Abdullah Reply

any genius online ? I need help!!

Guzorochi Reply

how can i help you?

Pina

need to learn polynomial

Zakariya

i will teach...

nandu

I'm waiting

Zakariya

plz help me in question

Abish

How can I help you?

Tlou

evaluate the following computation (x³-8/x-2)

Murtala Reply

teach me how to solve the first law of calculus.

Uncle Reply

teach me also how to solve the first law of calculus

Bilson

what is differentiation

Ibrahim Reply

only god knows😂

abdulkadir

f(x) = x-2 g(x) = 3x + 5 fog(x)? f(x)/g(x)

Naufal Reply

fog(x)= f(g(x)) = x-2 = 3x+5-2 = 3x+3 f(x)/g(x)= x-2/3x+5

diron

pweding paturo nsa calculus?

jimmy

how to use fundamental theorem to solve exponential

JULIA Reply

find the bounded area of the parabola y^2=4x and y=16x

Omar Reply

what is absolute value means?

Geo Reply

Chicken nuggets

Hugh

🐔

🐔🦃 nuggets

(mathematics) For a complex number a+bi, the principal square root of the sum of the squares of its real and imaginary parts, √a2+b2 . Denoted by | |. The absolute value |x| of a real number x is √x2 , which is equal to x if x is non-negative, and −x if x is negative.

Ismael

find integration of loge x

Game Reply

find the volume of a solid about the y-axis, x=0, x=1, y=0, y=7+x^3

Godwin Reply

how does this work

Brad Reply

Can calculus give the answers as same as other methods give in basic classes while solving the numericals?

Cosmos Reply

log tan (x/4+x/2)

Rohan

please answer

Rohan

y=(x^2 + 3x).(eipix)

Claudia

is this a answer

Ismael

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!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

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

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College algebra' conversation and receive update notifications?

Ask

	5 The point-slope formula By OpenStax Read Online Course
	3 Finding a linear equation By OpenStax Read Online Course
©flickr: Jovial	Video Mid Term By Dewey Compton Start Exam
	3 Sociology 03 Culture MCQ By OpenStax Start Quiz
	5 Lec:5 Case-control Studies By Janet Forrester Start Quiz
	Spreadsheets MCQ By Ryan Lowe Start Quiz
	22 Muscle and Pancreas Bio Path quiz By Brooke Delaney Start Exam
	23 AP Key Terms 23 The Digestive System By OpenStax Start Key Terms
	1 Gastrointestinal Pathophysiology By Laurence Bailen Start Exam
	6 Enterprise JavaBeans By JavaChamp Team Start Quiz
	2 Neuroanatomy 02 Brain External Features By Stephen Voron Start Quiz
	Chemistry Final By Briana Hamilton Start Flashcards