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dusya [7]
2 years ago
11

Find the equation of the line using the point-slope formula. Write the final equation using slope-intercept form. Perpendicular

to 5y=x-4 and passes through the point (-2,2)
Mathematics
1 answer:
polet [3.4K]2 years ago
6 0

Answer:

<u>Slope-intercept form</u>: y = -5x - 8

<u>Point-slope form</u>:  y - 2 = -5(x + 2)

Step-by-step explanation:

Given the equation, 5y = x - 4, which passes through point (-2, 2):

Transform the given equation into its <u>slope-intercept form</u>, y = mx + b.

In order to do so, divide both sides by 5 to isolate y:

5y = x - 4

\displaystyle\mathsf{\frac{5y}{5}\:=\:\frac{1x\:-\:4}{5}}

\displaystyle\mathsf{y\:=\:\frac{1}{5}x\:-\:\frac{4}{5}}  ⇒  This is the slope-intercept form of 5y = x - 4.

Next, we must determine the equation of the line that is perpendicular to \displaystyle\mathsf{y\:=\:\frac{1}{5}x\:-\:\frac{4}{5}}.    

<h2>Definition of Perpendicular Lines:</h2>

<u>Perpendicular lines</u> have <em>negative reciprocal</em> slopes.  This means that if we multiply the slopes of two lines, their product will equal to -1.  

In other words, if the slope of the given equation is m₁, and the slope of the other line perpendicular to the given linear equation is m₂, then:  m₁ × m₂ = -1.

  • Slope of the given equation: m₁ = ⅕
  • \displaystyle\mathsf{Slope\:of\:other\:line\:(m_2 )\:=\:-5\:or\:-\frac{5}{1}}

If we multiply these two slopes:

  • m₁ × m₂ = -1
  • \displaystyle\mathsf{m_1\:\times\\\:m_2\:=\:\frac{1}{5}\times\\-\frac{5}{1}\:=\:-1}

Now that we have identified the slope of the other line that is perpendicular to  5y = x - 4, we must determine the y-intercept of the <u>other line</u>.  

  • The <u>y-intercept</u> is the point on the graph where it crosses the y-axis, for which it is the value of "y" when its corresponding x-coordinate equals to zero (0).
  • Thus, the standard coordinates of the y-intercept is (0, <em>b</em>), for which its y-coordinate is the value of "<em>b</em>" in the slope-intercept form, y = mx + b.

Using the <u>slope</u> of the other line, m₂ = -5, and the given point, (-2, 2), substitute these values into the slope-intercept form to find the value of the y-intercept, <em>b</em>:

y = mx + b

2 = -5(-2) + b

2 = 10 + b

Subtract 10 from both sides to isolate b:

2 - 10 = 10 - 10 + b

-8 = b

The equation of the other line that is perpendicular to 5y = x - 4 is:

Linear Equation that is perpendicular to 5y = x - 4 in slope-intercept form:  

<h3>⇒   y = -5x - 8 </h3>

<h2>Rewrite the Equation in Point-slope Form:</h2>

The <u>point-slope form</u> is: y - y₁ = m(x - x₁)

In order to rewrite y = -5x - 8 in its point-slope form, we must substitute the value of the given point, (-2, 2) into the point-slope form:

y - y₁ = m(x - x₁)

y - 2 = -5[x - (-2)]

y - 2 = -5(x + 2) ⇒  This is the <u>point-slope form</u> of the line that is perpendicular to 5y = x - 4.

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Answer:

<h3><u>Mean</u></h3>

<u />

\textsf{Mean}\:\overline{X}=\sf \dfrac{\textsf{sum of all the data values}}{\textsf{total number of data values}}

\implies \sf Mean\:(Nilo)=\dfrac{5+6+14+15}{4}=\dfrac{40}{4}=10

\implies \sf Mean\:(Lisa)=\dfrac{8+9+11+12}{4}=\dfrac{40}{4}=10

<h3><u>Standard Deviation</u></h3>

\displaystyle \textsf{Standard Deviation }s=\sqrt{\dfrac{\sum X^2-\dfrac{(\sum X)^2}{n}}{n-1}}

\begin{aligned}\displaystyle \textsf{Standard Deviation (Nilo)} & =\sqrt{\dfrac{(5^2+6^2+14^2+15^2)-\dfrac{(5+6+14+15)^2}{4}}{4-1}}\\\\& = \sqrt{\dfrac{482-\dfrac{40^2}{4}}{3}}\\\\& = \sqrt{\dfrac{82}{3}}\\\\& = 5.23\end{aligned}

\begin{aligned}\displaystyle \textsf{Standard Deviation (Lisa)} & =\sqrt{\dfrac{(8^2+9^2+11^2+12^2)-\dfrac{(8+9+11+12)^2}{4}}{4-1}}\\\\& = \sqrt{\dfrac{410-\dfrac{40^2}{4}}{3}}\\\\& = \sqrt{\dfrac{10}{3}}\\\\& = 1.83\end{aligned}

<h3><u>Summary</u></h3>

Nilo has a mean score of 10 and a standard deviation of 5.23.

Lisa has a mean score of 10 and a standard deviation of 1.83.

The <u>mean</u> scores are the <u>same</u>.

Nilo's standard deviation is higher than Lisa's.  Therefore, Nilo's test scores are more <u>spread out</u> that Lisa's, which means Lisa's test scores are more <u>consistent</u>.

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