Question 673·Hard·Algebra

0 / 700

Line $L$ passes through the points $(a+1,\;2a-3)$ and $(4a-5,\;-6)$ , where $a$ is a real number. If line $L$ is perpendicular to the line with equation $5x + 2y = 7$ , what is the value of $a$ ?

The word perpendicular signals a negative reciprocal relationship between line slopes. When a line’s points contain an unknown, find its slope as the change in $y$ divided by the change in $x$ . Set that equal to the needed perpendicular slope, then use a Desmos regression to solve. Flip the given slope, not the unknown you’re asked to find.

Hints

Hint 1
For $Ax+By=C$ , the slope is $-A/B$ : moving right by $B$ makes $y$ move down by $A$ . What slope comes from the given equation, and what negative reciprocal must line $L$ have?
Hint 2
The slope between two points is the change in $y$ divided by the change in $x$ . Subtract the coordinates in the same order on top and bottom. What expression does that give in terms of $a$ ?
Hint 3
Both calculations describe line $L$ ’s slope, so they must be equal. Turn that equality into a one-unknown regression in Desmos. What value does Desmos find for $a$ ?

Step-by-step

Match the perpendicular slope to the slope from the points
Step 1
Find the given line’s slope
Type $5x+2y=7$ in Desmos; its graph falls to the right. For $Ax+By=C$ , the slope (vertical change divided by horizontal change) is $-A/B$ , so this line’s slope is $-5/2$ . The $7$ doesn’t affect its slope.
Step 2
Get the slope line L needs
Because the lines are perpendicular, flip the fraction and change the sign of the given slope. Line $L$ needs the negative reciprocal of $-5/2$ :
$m_L=\frac25.$
Step 3
Find line L’s slope from its points
Subtract the first point from the second on top and bottom. The slope is vertical change over horizontal change:
$m_L=\frac{-6-(2a-3)}{(4a-5)-(a+1)}.$
Simplify the top:
$m_L=\frac{-2a-3}{(4a-5)-(a+1)}.$
Simplify the bottom:
$m_L=\frac{-2a-3}{3a-6}.$
The bottom can’t be $0$ : line $L$ needs the finite slope $2/5$ , not a vertical line.
Step 4
Set the two slopes equal
Both expressions give the same slope, so match the slope from the points to the perpendicular slope:
$\frac{-2a-3}{3a-6}=\frac25.$
Don’t use $-5/2$ on the right. That belongs to the given line, not line $L$ .
Step 5
Solve for a in Desmos
Enter $\frac{-2a-3}{3a-6}\sim\frac25$ . The regression symbol $\sim$ asks Desmos to find $a$ that makes the sides match. Under PARAMETERS, Desmos shows $a=-0.1875$ . Type $a$ on the next line and tap its fraction button to get $-\frac3{16}$ . That’s the value of $a$ . Choice C.

Question 673·Hard·Algebra

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this