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kakasveta [241]
3 years ago
12

A line goes through the points (9,8) and (4,-12). What is the slope of the line? Show your work Write the equation of the line i

n point-slope form. Show your work Write the equation of the line in slope-intercept form. Show your work.
Mathematics
1 answer:
RoseWind [281]3 years ago
5 0

The slope of the line is 4

The point slope form of the line can be written as (y - 8) = 4(x - 9)

The slope intercept form of the line is y = 4x - 28

In order to find the slope you will need to use the formula for slope. It is included below.

m(slope) = (y2 - y1)/(x2 - x1)

m = (8 - -12)/(9 - 4)

m = (8 + 12)/(9 - 4)

m = 20/5

m = 4

Now that we have the slope, we can plug it into point slope form along with one of the points.

(y - y1) = m(x - x1)

(y - 8) = 4(x - 9)

Now to get to the slope-intercept form of the line, we can simply manipulate the point-slope form until it is in y = mx + b

(y - 8) = 4(x - 9) ----> Distribute the 4

y - 8 = 4x - 36 -----> Add 8 to both sides

y = 4x - 28

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A. Factor the numerator as a difference of squares:

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c. As x\to\infty, the contribution of the terms of degree less than 2 becomes negligible, which means we can write

\displaystyle\lim_{x\to\infty}\frac{4x^2-4x-8}{x^2-9}=\lim_{x\to\infty}\frac{4x^2}{x^2}=\lim_{x\to\infty}4=4

e. Let's first rewrite the root terms with rational exponents:

\displaystyle\lim_{x\to1}\frac{\sqrt[3]x-x}{\sqrt x-x}=\lim_{x\to1}\frac{x^{1/3}-x}{x^{1/2}-x}

Next we rationalize the numerator and denominator. We do so by recalling

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(a-b)(a^2+ab+b^2)=a^3-b^3

In particular,

(x^{1/3}-x)(x^{2/3}+x^{4/3}+x^2)=x-x^3
(x^{1/2}-x)(x^{1/2}+x)=x-x^2

so we have

\displaystyle\lim_{x\to1}\frac{x^{1/3}-x}{x^{1/2}-x}\cdot\frac{x^{2/3}+x^{4/3}+x^2}{x^{2/3}+x^{4/3}+x^2}\cdot\frac{x^{1/2}+x}{x^{1/2}+x}=\lim_{x\to1}\frac{x-x^3}{x-x^2}\cdot\frac{x^{1/2}+x}{x^{2/3}+x^{4/3}+x^2}

For x\neq0 and x\neq1, we can simplify the first term:

\dfrac{x-x^3}{x-x^2}=\dfrac{x(1-x^2)}{x(1-x)}=\dfrac{x(1-x)(1+x)}{x(1-x)}=1+x

So our limit becomes

\displaystyle\lim_{x\to1}\frac{(1+x)(x^{1/2}+x)}{x^{2/3}+x^{4/3}+x^2}=\frac{(1+1)(1+1)}{1+1+1}=\frac43
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I would say 80, because without an estimate it'd be 79 and 79x5=395. So 80x5=400, which is the estimate of 395. So 80 is the answer
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