Question 299·Hard·Algebra

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The graph shows lines $m$ and $n$ in the $xy$ -plane.

Line $m$ is defined by the equation $5x-2y=4$ . Line $n$ passes through the points $P$ and $Q$ shown on the graph.

Line $p$ is perpendicular to line $n$ and passes through the midpoint of segment $\overline{PQ}$ . If the solution to the system consisting of lines $m$ and $p$ is $(x,y)$ , which choice is the value of $12y-6x$ ?

The cue here is a line through a segment’s midpoint that’s perpendicular to the segment’s line. That line is a perpendicular bisector. Read the endpoints, find the midpoint, and use the negative reciprocal of the segment line’s slope to write its equation. Then graph it with the given line in Desmos. Intersecting the original segment line instead is the trap.

Hints

Hint 1
A line’s slope is its change in $y$ divided by its change in $x$ . Read the coordinates of $P$ and $Q$ from the grid, then subtract in the same order on top and bottom.
Hint 2
The midpoint is halfway between the endpoints. Average their $x$ -coordinates to get its $x$ -coordinate, and average their $y$ -coordinates to get its $y$ -coordinate. Don’t swap coordinates between the two averages.
Hint 3
Perpendicular lines have slopes that are negative reciprocals: flip the fraction and change its sign. Use that new slope with the midpoint to write line $p$ , then find where $p$ meets line $m$ .

Step-by-step

Build line $p$ and graph the system
Step 1
Find the slope of line $n$
The graph marks $P=(0,8)$ and $Q=(8,2)$ . Slope is the change in $y$ divided by the change in $x$ , so subtract in the same order: $s_n=\frac{2-8}{8-0}=-\frac34$ . Type $\frac{2-8}{8-0}$ in Desmos. It shows $-0.75$ , or $-\frac34$ with the fraction button.
Step 2
Find the midpoint of $PQ$
The midpoint is halfway between $P$ and $Q$ , so average each coordinate separately. Type $\frac{0+8}{2}$ and $\frac{8+2}{2}$ in Desmos. It shows $4$ and $5$ , giving the midpoint $(4,5)$ .
Step 3
Get the perpendicular slope
Line $p$ is perpendicular to line $n$ . Perpendicular slopes are negative reciprocals: flip the fraction and change the sign. Starting from $-\frac34$ , that gives line $p$ the positive slope $\frac43$ . Keeping $-\frac34$ would give you line $n$ again.
Step 4
Write line $p$ through the midpoint
Use the slope $\frac43$ and midpoint $(4,5)$ . In point-slope form, $y-5$ is the vertical change from that point and $x-4$ is the horizontal change, so line $p$ is: $y-5=\frac43(x-4)$ . This line has both the required slope and the required midpoint.
Step 5
Find where lines $m$ and $p$ meet
Type $5x-2y=4$ and $y-5=\frac43(x-4)$ on separate Desmos lines. Click their intersection. Desmos labels the point approximately $(1.429,1.571)$ ; its first coordinate is $x$ , and its second is $y$ .
Step 6
Evaluate the requested expression
The question asks for $12y-6x$ , not either coordinate alone. Type $12(1.571)-6(1.429)$ in Desmos; it shows $10.278$ . Then type $\frac{72}{7}$ ; Desmos shows about $10.286$ . The small difference comes from the rounded intersection coordinates, so the matching value is $\frac{72}{7}$ . Choice A.

Question 299·Hard·Algebra

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