Question 50·200 Super-Hard SAT Math Questions·Algebra

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In the given pair of equations, $m$ and $n$ are constants. The graph of this pair of equations in the $xy$ -plane is a pair of perpendicular lines. Which choice of the following pairs of equations also represents a pair of perpendicular lines?

15x - 26y = 4

mx + ny = 4

For lines in standard form, the $x$ - and $y$ -coefficients give you the slope. If the lines are perpendicular, their slopes multiply to $-1$ . Find what that tells you about the unknown coefficients, then choose one convenient pair of values with that ratio and check the answer choices in Desmos. Watch for a choice that scales only one coefficient: that changes the slope.

Hints

Hint 1
For $Ax+By=C$ , the slope is $-A/B$ . The sign on the $y$ -coefficient matters: what slope does $15x-26y=4$ have?
Hint 2
Perpendicular slopes are negative reciprocals: flip the fraction and change its sign. Set the slope of $mx+ny=4$ equal to the negative reciprocal of the first slope. What does that tell you about $m/n$ ?
Hint 3
Once you know $m/n$ , choose convenient values of $m$ and $n$ with that ratio. For each answer pair, multiply its two slopes. Which product must equal $-1$ ?

Step-by-step

Find the coefficient ratio, then test slope products
Step 1
Find the first line’s slope
For a line written $Ax+By=C$ , its slope is $-A/B$ . The $y$ -coefficient in $15x-26y=4$ is $-26$ , not $26$ , so its slope is:
$-\frac{15}{-26}=\frac{15}{26}.$
Step 2
Use perpendicularity to relate $m$ and $n$
The slope of $mx+ny=4$ is $-m/n$ . Perpendicular slopes are negative reciprocals, so the slope perpendicular to $15/26$ is $-26/15$ . Set those equal:
$-\frac{m}{n}=-\frac{26}{15}.$
Cancel the minus signs:
$\frac{m}{n}=\frac{26}{15}.$
Both coefficients are nonzero: neither of the given perpendicular lines is horizontal or vertical.
Step 3
Choose values that fit the ratio
Only the ratio $m/n$ matters for these slopes, not the individual values. Choose $m=26$ and $n=15$ . Type $m=26$ , $n=15$ , and $\frac{15}{26}(-m/n)$ in Desmos. It prints $-1$ , confirming that this valid pair of values gives the stated perpendicular lines.
Step 4
Test the answer pairs
For each pair, multiply its slopes. For example, $nx-my=1$ has slope $-n/(-m)=n/m$ ; keep that minus sign on $my$ . Add these four products in answer-choice order. Desmos prints $1$ , $-1$ , $-2$ , and $-0.5$ :
$\begin{aligned} \left(-\frac{26}{15}\right)\left(-\frac{n}{m}\right)&=1\\[1.4em] \left(-\frac{26}{15}\right)\left(\frac{n}{m}\right)&=-1\\[1.4em] \left(\frac{15}{26}\right)\left(-\frac{2m}{n}\right)&=-2\\[1.4em] \left(\frac{30}{52}\right)\left(-\frac{m}{2n}\right)&=-0.5 \end{aligned}$
Only the second product is $-1$ , so $26x+15y=1$ and $nx-my=1$ are perpendicular. Choice B.

Question 50·200 Super-Hard SAT Math Questions·Algebra

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