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

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

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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 $x = −1.$

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

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

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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 = \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 $- \frac{4}{7} .$

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

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

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

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 $b = −4.$

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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.

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}

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Questions & Answers

how did you get 1640

Noor Reply

If auger is pair are the roots of equation x2+5x-3=0

Peter Reply

Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.

MATTHEW Reply

420

Sharon

from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph

George

Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28

Salma

Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.

Aaron Reply

find product (-6m+6) ( 3m²+4m-3)

SIMRAN Reply

-42m²+60m-18

Salma

what is the solution

bill

how did you arrive at this answer?

bill

-24m+3+3mÁ^2

Susan

i really want to learn

Amira

I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please

Amira

Hw did u arrive to this answer.

Aphelele

Bajemah

-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18

Salma

complete the table of valuesfor each given equatio then graph. 1.x+2y=3

Jovelyn Reply

x=3-2y

Salma

y=x+3/2

Salma

Enock

given that (7x-5):(2+4x)=8:7find the value of x

Nandala

3x-12y=18

Kelvin

please why isn't that the 0is in ten thousand place

Grace Reply

please why is it that the 0is in the place of ten thousand

Grace

Send the example to me here and let me see

Stephen

A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg

Marry Reply

how far

Abubakar

cool u

Enock

state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°

Abegail Reply

hello

BenJay

Method

I am eliacin, I need your help in maths

Rood

how can I help

Sir

hmm can we speak here?

Amoon

however, may I ask you some questions about Algarba?

Amoon

Enock

what the last part of the problem mean?

Roger

The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.

cameron Reply

Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.

mahnoor Reply

I'm guessing, but it's somewhere around $4335.00 I think

Lewis

12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00

Munster

difference between rational and irrational numbers

Arundhati Reply

When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?

Jakoiya Reply

how to reduced echelon form

Solomon Reply

Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?

Zack Reply

d=r×t the equation would be 8/r+24/r+4=3 worked out

Sheirtina

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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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