Question 533·Medium·Algebra

0 / 700

In the $xy$ -plane, line $p$ is perpendicular to the line with equation $y = -\frac{1}{2}x + 6$ and passes through the point $(4,-3)$ . What is the $y$ -intercept of line $p$ (the value of $b$ in $y = mx + b$ )?

A line written as $y=mx+b$ shows its slope, the change in $y$ for each increase of $1$ in $x$ . The word perpendicular tells you to flip that slope’s fraction and change its sign. Use the new slope with the given point to find the intercept. Keeping the old slope would give a parallel line instead.

Hints

Hint 1
In slope-intercept form $y=mx+b$ , the number multiplying $x$ is the slope. Read that number from the given equation before working with line $p$ .
Hint 2
For perpendicular lines, the slopes multiply to $-1$ . Flip the given slope’s fraction and change its sign. What slope does that give line $p$ ?
Hint 3
The $y$ -intercept is the value of $y$ when $x=0$ . The given point has a different $x$ -coordinate, so its $y$ -coordinate isn’t the intercept. Put the point and your new slope into $y=mx+b$ .

Step-by-step

Find the perpendicular slope, then the intercept
Step 1
Read the given line’s slope
In slope-intercept form $y=mx+b$ , $m$ is the slope. The given equation has slope $-\frac{1}{2}$ . Its $6$ is the given line’s intercept, not line $p$ ’s.
Step 2
Find line $p$ ’s slope
Slopes of perpendicular lines multiply to $-1$ . Flip $-\frac{1}{2}$ to $-2$ , then change its sign, so line $p$ has slope $2$ . Check both changes: $\left(-\frac{1}{2}\right)(2)=-1$ . Changing only the sign or only flipping the fraction won’t make that product $-1$ .
Step 3
Put the point into line $p$ ’s equation
The point $(4,-3)$ says that $y=-3$ when $x=4$ . Use those coordinates with line $p$ ’s slope, $2$ , in $y=2x+b$ :
$-3=2(4)+b$
The $-3$ isn’t the intercept: at a $y$ -intercept, $x=0$ , not $4$ .
Step 4
Solve for the intercept in Desmos
Type $-3\sim 2(4)+b$ . The $\sim$ tells Desmos to find the unknown $b$ ; under PARAMETERS, it shows $b=-11$ . So line $p$ crosses the $y$ -axis at $(0,-11)$ , and the requested intercept value is $-11$ . Choice D.

Question 533·Medium·Algebra

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this