Question 553·Hard·Algebra

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Consider the following system of linear equations.

\begin{aligned} 5x + k y &= 10 \\[1.4em] 2x - 5y &= 7 \end{aligned}

If the graphs of the two equations are perpendicular in the $xy$ -plane, what is the value of $k$ ?

Your answer

The word perpendicular signals a negative reciprocal slope relationship, even when the lines are given as equations rather than drawn. For a line written as $Ax+By=C$ , use $-A/B$ to find its slope. Multiply the two slopes, set their product to $-1$ , and use a Desmos regression to find the missing coefficient. Setting the slopes equal would test for parallel lines instead.

Hints

Hint 1
A line’s slope tells you its rise divided by its run. In $Ax+By=C$ , you can read it as $-A/B$ when $B$ isn’t zero. Keep the negative sign in the second equation’s $y$ coefficient.
Hint 2
Perpendicular slopes are negative reciprocals, so they multiply to $-1$ , not $1$ . Write a slope for each line, then use that product to make an equation with only $k$ unknown.

Step-by-step

Multiply the perpendicular slopes
Step 1
Find the slope of the second line
For $Ax+By=C$ , the slope (rise divided by run) is $-A/B$ . In $2x-5y=7$ , the $x$ coefficient is $2$ and the $y$ coefficient is $-5$ , so:
$m_2=-\frac{2}{-5}.$
Simplify the two negative signs:
$m_2=\frac25.$
Step 2
Find the slope containing $k$
In $5x+ky=10$ , the same rule gives: $m_1=-\frac5k$ .Here $k$ can’t be $0$ : that would make this line vertical, but a line perpendicular to it would have to be horizontal. The second line has slope $\frac25$ , so it isn’t horizontal.
Step 3
Turn perpendicular into an equation
Perpendicular slopes multiply to $-1$ , not $1$ . They’re negative reciprocals: turning a line through a right angle flips its slope fraction and changes its sign. Use the two slopes you found:
$\left(-\frac5k\right)\left(\frac25\right)=-1.$
Step 4
Solve for the missing coefficient
Type $\left(-\frac{5}{k_1}\right)\left(\frac25\right)\sim-1$ in Desmos. Use $k_1$ for the unknown $k$ and $\sim$ instead of $=$ so Desmos fits its value rather than making a slider. Under PARAMETERS, Desmos shows $k_1=2$ . So the coefficient that makes the lines perpendicular is $k=2$ . Grid in 2.

Lessons that teach this

SAT AlgebraIntermediateCoreWork with parallel and perpendicular lines

Question 553·Hard·Algebra

Strategy

Hints

Step-by-step

Lessons that teach this

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Strategy

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Step-by-step

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