The line that passes through the points and in the -plane is perpendicular to the line . What is the value of ?

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SAT Linear Equations in Two Variables Question 96 | Free SAT Prep | Aniko

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Question 96·Hard·Linear Equations in Two Variables

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The line that passes through the points $(k,3)$ and $(2k,-1)$ in the $xy$ -plane is perpendicular to the line $4x + 5y = 7$ . What is the value of $k$ ?

For perpendicular-line questions, first convert any standard-form equation to slope-intercept form to read off the slope quickly. Use the fact that perpendicular slopes are negative reciprocals (their product is -1) to find the needed slope, then compute the slope between the given points using $(y_2 - y_1)/(x_2 - x_1)$ and set it equal to that perpendicular slope. Solve the resulting simple equation in one variable carefully, watching signs when cross-multiplying.

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Start with the given line

Rewrite $4x + 5y = 7$ in the form $y = mx + b$ so you can clearly see its slope.

Use the perpendicular relationship

For two perpendicular lines, how are their slopes related? Think about the product of their slopes.

Find the slope through the points with k

Use the slope formula with points $(k,3)$ and $(2k,-1)$ to express the slope of that line in terms of $k$ .

Set up an equation with the slopes

Set the slope you found from the points equal to the slope required for a line perpendicular to the given line, then solve that equation for $k$ .

Desmos Guide

Find the slope of the given line

In Desmos, type y = (-4/5)*x + 7/5 to graph the line $4x + 5y = 7$ . Note its slope is $-4/5$ , so the perpendicular slope is $5/4$ .

Express the slope from the two points

The slope through $(k,3)$ and $(2k,-1)$ is $(-1-3)/(2k-k) = -4/k$ . Type y = -4/x and y = 5/4 in Desmos.

Find the intersection

Click on the intersection of y = -4/x and y = 5/4. The x-coordinate of this intersection is the value of $k$ .

Step-by-step Explanation

Find the slope of the given line

The given line is

4x + 5y = 7.

Rewrite it in slope-intercept form $y = mx + b$ to see its slope:

\begin{aligned} 5y &= -4x + 7 \\ y &= -\frac{4}{5}x + \frac{7}{5}. \end{aligned}

So the slope of this line is $m = -\frac{4}{5}$ .

Determine the slope of a perpendicular line

If two lines are perpendicular, their slopes are negative reciprocals. That means if one slope is $m$ , the other is $-\dfrac{1}{m}$ .

Here, the given slope is $-\dfrac{4}{5}$ , so the slope of any line perpendicular to it must satisfy

\left(-\frac{4}{5}\right) \cdot m_{\perp} = -1.

Solving this gives $m_{\perp} = \dfrac{5}{4}$ (the negative reciprocal of $-\dfrac{4}{5}$ ).

Write the slope of the line through the two points

The line in the question passes through $(k,3)$ and $(2k,-1)$ .

The slope of a line through $(x_1,y_1)$ and $(x_2,y_2)$ is

\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}.

So the slope of this line is

\frac{-1 - 3}{2k - k} = \frac{-4}{k}.

Set the slopes equal and solve for k

Because this line is perpendicular to the given line, its slope $-\dfrac{4}{k}$ must equal $\dfrac{5}{4}$ :

-\frac{4}{k} = \frac{5}{4}.

Cross-multiply:

\begin{aligned} -4 \cdot 4 &= 5k \\ -16 &= 5k. \end{aligned}

Divide both sides by 5:

k = -\frac{16}{5}.

So the value of $k$ is $-\dfrac{16}{5}$ , which corresponds to choice A.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation