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 does the planets on our solar system orbit

cheten Reply

how many Messier objects are there in space

satish Reply

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Richard Reply

what are astronomy

Issan Reply

Astronomy (from Ancient Greek ἀστρονομία (astronomía) 'science that studies the laws of the stars') is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution.

Rafael

vjuvu

Elgoog

what is big bang theory?

Rosemary

what type of activity astronomer do?

Rosemary

Richard

the big bang theory is a theory which states that all matter was compressed together in one place the matter got so unstable it exploded releasing All its contents in the form of hydrogen

Roaul

I want to be an astronomer. That's my dream

Astrit

Who named the the whole galaxy?

Shola Reply

solar Univers

GPOWER

what is space

Richard

what is the dark matter

Richard

what are the factors upon which the atmosphere is stratified

Nicholas Reply

is the big bang the sun

Folakemi Reply

Sokak

bigbang is the beginning of the universe

Sokak

but thats just a theory

Sokak

nothing will happen, don't worry brother.

Vansh

what does comet means

GANGAIN Reply

these are Rocky substances between mars and jupiter

GANGAIN

Comets are cosmic snowballs of frozen gases , rock and dust that orbit the sun. They are mostly found between the orbits of Venus and Mercury.

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hey can anyone guide me abt international astronomy olympiad

sahil

how can we learn right and true ?

Govinda Reply

why the moon is always appear in an elliptical shape

Gatjuol Reply

Because when astroid hit the Earth then a piece of elliptical shape of the earth was separated which is now called moon.

Hemen

what's see level?

lidiya Reply

Did you mean eye sight or sea level

Minal

oh sorry it's sea level

lidiya

according to the theory of astronomers why the moon is always appear in an elliptical orbit?

Gatjuol

hi !!! I am new in astronomy.... I have so many questions in mind .... all of scientists of the word they just give opinion only. but they never think true or false ... i respect all of them... I believes whole universe depending on true ...থিউরি

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we're all stars and galaxies a part of sun. how can science prove thx with respect old ancient times picture or books..or anything with respect to present time .but we r a part of that universe

w astronomy and cosmology!

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another theory of universe except big ban

Albash Reply

how was universe born

Asmit Reply

there many theory to born universe but what is the reality of big bang theory to born universe

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what is the exact value of π?

Nagalakshmi

by big bang

universal

there are many theories regarding this it's on you believe any theory that you think is true ex. eternal inflation theory, oscillation model theory, multiple universe theory the big bang theory etc.

Aarya

I think after Big Bang!

Michele

from where on earth could u observe all the stars during the during the course of an year

Karuna Reply

I think it couldn't possible on earth

Nagalakshmi

in this time i don't Know

Michele

is that so. the question was in the end of this chapter

Karuna

in theory, you could see them all from the equator (though over the course of a year, not at pne time). stars are measured in "declination", which is how far N or S of the equator (90* to -90*). Polaris is the North star, and is ALMOST 90* (+89*). So it would just barely creep over the horizon.

Christopher

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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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