In the -plane, line passes through the points and . Line is perpendicular to line and passes through the midpoint of the segment with endpoints and . What is the -intercept of line ?

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SAT Linear Equations in Two Variables Question 90 | Free SAT Prep | Aniko

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Question 90·Hard·Linear Equations in Two Variables

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In the $xy$ -plane, line $q$ passes through the points $(4,-3)$ and $(10,7)$ . Line $s$ is perpendicular to line $q$ and passes through the midpoint of the segment with endpoints $(4,-3)$ and $(10,7)$ . What is the $y$ -intercept of line $s$ ?

For perpendicular-line questions, move systematically: first, compute the slope of the given line using the slope formula; second, take the negative reciprocal for the perpendicular slope; third, find any required point using midpoint, distance, or coordinates given; and finally plug the perpendicular slope and that point into point-slope form and rearrange to slope-intercept form so you can quickly read the $y$ -intercept or other requested value. Keeping your fractions organized and converting whole numbers to fractions (like $2 = 10/5$ ) helps avoid small arithmetic mistakes that cost points.

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Start with the slope of line $q$

Use the two given points on line $q$ and apply the slope formula $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ .

Use the perpendicular relationship

Once you know the slope of line $q$ , think about how the slope of a perpendicular line is related to it. What does "negative reciprocal" mean?

Find the point line $s$ must pass through

Line $s$ passes through the midpoint of the segment connecting $(4,-3)$ and $(10,7)$ . Use the midpoint formula $\left( \dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2} \right)$ .

Use point-slope form to get the $y$ -intercept

With the slope of $s$ and the midpoint, write the equation in point-slope form $y - y_1 = m(x - x_1)$ , then rearrange into $y = mx + b$ and read off the $y$ -intercept $b$ .

Desmos Guide

Graph line $q$ and find its slope

Enter the points $(4,-3)$ and $(10,7)$ in Desmos (for example, as a table or using the point feature), then use the slope formula manually or let Desmos fit a line through them (e.g., type y1 ~ m x1 + b) and note the slope $m$ shown.

Determine the perpendicular slope

Take the slope you found for line $q$ and compute its negative reciprocal (you can type the fraction into Desmos as an expression like -3/5 to verify the value).

Compute the midpoint in Desmos

Type the expressions (4+10)/2 and (-3+7)/2 into Desmos separately to confirm the midpoint coordinates that line $s$ must pass through.

Write line $s$ and read the $y$ -intercept

In Desmos, enter the equation of line $s$ in point-slope form using the perpendicular slope and the midpoint, like y - 2 = (-3/5)(x - 7), then let Desmos rewrite it in $y = mx + b$ form or click on the graph where it crosses the $y$ -axis ( $x=0$ ) to read off the $y$ -intercept.

Step-by-step Explanation

Find the slope of line $q$

Line $q$ passes through $(4,-3)$ and $(10,7)$ . Use the slope formula:

m_q = \frac{7 - (-3)}{10 - 4} = \frac{7 + 3}{6} = \frac{10}{6} = \frac{5}{3}.

So the slope of line $q$ is $5/3$ .

Find the slope of the perpendicular line $s$

Line $s$ is perpendicular to line $q$ . Perpendicular lines have slopes that are negative reciprocals of each other.

Slope of $q$ is $5/3$ .
The negative reciprocal of $5/3$ is $-3/5$ .

So the slope of line $s$ is $-3/5$ .

Find the midpoint of the segment

Line $s$ passes through the midpoint of the segment with endpoints $(4,-3)$ and $(10,7)$ .

Use the midpoint formula:

\left( \frac{4+10}{2}, \frac{-3+7}{2} \right) = \left( \frac{14}{2}, \frac{4}{2} \right) = (7,2).

So line $s$ goes through the point $(7,2)$ .

Write the equation of line $s$ and find its $y$ -intercept

Use the point-slope form with slope $-3/5$ and point $(7,2)$ :

y - 2 = -\frac{3}{5}(x - 7).

Distribute $-3/5$ :

y - 2 = -\frac{3}{5}x + \frac{21}{5}.

Add $2$ (which is $10/5$ ) to both sides to solve for $y$ :

\begin{aligned} y &= -\frac{3}{5}x + \frac{21}{5} + 2 \\ &= -\frac{3}{5}x + \frac{21}{5} + \frac{10}{5} \\ &= -\frac{3}{5}x + \frac{31}{5}. \end{aligned}

The $y$ -intercept is the constant term when the equation is in the form $y = mx + b$ , so the $y$ -intercept is $\dfrac{31}{5}$ .

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Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation