Question 478·Hard·Algebra

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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 $k$ ?

With perpendicular lines, the two slopes multiply to $-1$ . Read the slope of the given equation, flip the fraction and change its sign, then find the slope through the two points. Set those slopes equal and use a Desmos regression to find the missing coordinate. The trap is subtracting the point coordinates in different orders, which reverses the slope's sign.

Hints

Hint 1
For a line written $Ax+By=C$ , the slope is $-A/B$ : those are the numbers multiplying $x$ and $y$ . Which coefficients play the roles of $A$ and $B$ in the given equation?
Hint 2
Perpendicular lines have slopes that multiply to $-1$ . Once you know the given line's slope, flip its fraction and change its sign. What slope must the line through the two points have?
Hint 3
A slope is the change in $y$ divided by the change in $x$ . Subtract the coordinates in the same order above and below the fraction, then match that slope to the one you need.

Step-by-step

Match the perpendicular slope to the point-pair slope
Step 1
Read the given line's slope
For a line in standard form, $Ax+By=C$ , the slope is $-A/B$ : moving the $x$ term across and dividing by $B$ makes $-A/B$ the coefficient of $x$ . The coefficients here are $4$ and $5$ , so the given line's slope is $-\frac45$ . The $7$ changes where the line sits, not its slope.
Step 2
Find the slope of a perpendicular line
Because the lines are perpendicular, their slopes must multiply to $-1$ . Take the negative reciprocal of $-\frac45$ : flip the fraction and change its sign to get $\frac54$ . A perpendicular slope needs both the flip and the sign change.
Step 3
Find the slope through the two points
A slope is the change in $y$ over the change in $x$ . Subtract the second point minus the first in the same order on top and bottom:
$m=\frac{-1-3}{2k-k}.$
Simplify the differences:
$m=\frac{-4}{k}.$
The change in $x$ is $k$ , not $2k$ , because you subtract the first point's $x$ -coordinate.
Step 4
Match the slopes and find the coordinate
The slope from the points must equal the perpendicular slope:
$\frac{-4}{k}=\frac54.$
Type $\frac{-4}{k}\sim\frac54$ in Desmos. The $\sim$ makes a regression: Desmos finds the unknown $k$ and shows $k=-3.2$ under PARAMETERS. Type $k$ on the next line and click its fraction button to see $-\frac{16}{5}$ . So the value of $k$ is $-\frac{16}{5}$ . Choice D.

Question 478·Hard·Algebra

Strategy

Hints

Step-by-step

Lessons that teach this

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Strategy

Hints

Step-by-step

Lessons that teach this