Question 613·Hard·Algebra

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In the $xy$ -plane, a line $\ell$ passes through the point $(4,\,k)$ and is perpendicular to the line $3x - 2y = 7$ . The $x$ -intercept of $\ell$ is $(k+1,\,0)$ . What is the value of $k$ ?

Your answer

For perpendicular lines, which meet at a right angle, the slopes are negative reciprocals: flip the fraction and change its sign. Read the given line’s slope from its equation, then use the two points on the new line to write another slope. Set that slope equal to the perpendicular slope and use a Desmos regression to solve. Don't miss the sign change.

Hints

Hint 1
In standard form $Ax+By=C$ , the slope is $-A/B$ when $B$ isn’t $0$ . Use both coefficients, including their signs, to find the slope of the given line.
Hint 2
For perpendicular lines, the slopes multiply to $-1$ . Flip the fraction and change its sign to get the slope the new line needs.
Hint 3
An $x$ -intercept is a point where $y=0$ . You have two points on the same line, so write its slope as the change in $y$ divided by the matching change in $x$ .

Step-by-step

Match the perpendicular slope to the slope from two points
Step 1
Find the given line’s slope
Type $3x-2y=7$ in Desmos. Its graph rises as $x$ increases, so expect a positive slope. For a line in standard form $Ax+By=C$ , the slope, or change in $y$ for each increase of $1$ in $x$ , is $-A/B$ . Here it is $-3/(-2)=\frac{3}{2}$ .
Step 2
Get the perpendicular slope
Line $\ell$ is perpendicular to that line, so its slope is the negative reciprocal of $\frac{3}{2}$ : flip the fraction and change its sign to get $-\frac{2}{3}$ .
Step 3
Write the slope using the two points
Both $(4,k)$ and the $x$ -intercept $(k+1,0)$ lie on $\ell$ . Using rise over run, subtract the coordinates in the same order: $m_\ell=\frac{0-k}{(k+1)-4}=\frac{-k}{k-3}$ . If $k=3$ , both points have $x=4$ , making $\ell$ vertical. It couldn’t be perpendicular to a line with slope $\frac{3}{2}$ .
Step 4
Match the two slopes
These are two ways to describe the slope of the same line. So set the slope from the points equal to the perpendicular slope:
$\frac{-k}{k-3}=-\frac{2}{3}.$
Step 5
Clear the fractions
Since $k$ cannot be $3$ , multiply both sides by $3(k-3)$ to get a linear equation:
$-3k=-2(k-3).$
Step 6
Solve for the requested value
Replace $k$ with $x_1$ and $=$ with $\sim$ so Desmos can fit the unknown value. Type $-3x_1\sim-2(x_1-3)$ below the earlier line. Desmos reports $x_1=-6$ under PARAMETERS, so $k=-6$ . Grid in -6.

Question 613·Hard·Algebra

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