Question 568·Hard·Algebra

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In the $xy$ -plane, line $m$ passes through the point $(-4,9)$ and is perpendicular to the line defined by $2x + 5y = 7$ . Line $m$ intersects the line $y = 3x - 2$ at the point $(p,q)$ . What is the value of $p + q$ ?

Your answer

A line’s slope tells you its direction, and a point tells you where it sits. If two lines are perpendicular, swap the slope’s numerator and denominator and change its sign to get the new slope. Use that slope and the given point to write line $m$ , then graph it with the other line in Desmos. The intersection may not be the final value requested.

Hints

Hint 1
For a line in standard form $Ax+By=C$ , the slope is $-A/B$ . Which numbers multiply $x$ and $y$ in the given equation? Keep the minus sign when you find its slope.
Hint 2
Perpendicular lines have negative reciprocal slopes: swap the top and bottom of the fraction and change its sign. Then use point-slope form, $y-y_1=s(x-x_1)$ , to make line $m$ pass through the given point.
Hint 3
An intersection lies on both lines. Graph line $m$ and $y=3x-2$ in Desmos, then click where they meet. Which coordinate is $p$ , which is $q$ , and what does the question ask you to do with them?

Step-by-step

Find the perpendicular line, then its intersection
Step 1
Find the given line’s slope
Write $2x+5y=7$ in slope-intercept form $y=sx+b$ , where $s$ is the slope. Subtract $2x$ from both sides:
$5y=-2x+7$
Divide every term by $5$ :
$y=-\frac25x+\frac75$
So the given line’s slope is $-\frac25$ .
Step 2
Find line $m$ ’s slope
For perpendicular lines, swap the slope’s numerator and denominator and change its sign. The negative reciprocal of $-\frac25$ is positive, so line $m$ has slope
$s_m=\frac52$
Using $-\frac52$ would swap the numbers but miss the sign change.
Step 3
Write line $m$ through the given point
Point-slope form $y-y_1=s(x-x_1)$ makes a line with slope $s$ pass through $(x_1,y_1)$ . Put in slope $\frac52$ and the point $(-4,9)$ :
$y-9=\frac52\bigl(x-(-4)\bigr)$
Replace the double negative with a plus sign:
$y-9=\frac52(x+4)$
Step 4
Read the intersection
Type $2x+5y=7$ , $y-9=\frac52(x+4)$ , and $y=3x-2$ on separate Desmos lines. The original line slopes down; click where the two rising lines meet. Desmos shows $(42,124)$ . That intersection belongs to line $m$ and $y=3x-2$ , so its first coordinate is $p=42$ and its second is $q=124$ .
Step 5
Add the coordinates
The question asks for the sum, not either coordinate alone. Type $42+124$ on a new Desmos line; it prints $166$ . So $p+q=166$ . Grid in 166.

Question 568·Hard·Algebra

Strategy

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Step-by-step

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