Question 262·Medium·Algebra

0 / 700

The line $\ell$ has a slope that is twice the slope of the line through the points $(1,2)$ and $(5,-2)$ . If line $\ell$ also passes through the point $(4,-3)$ , what is the $y$ -intercept of line $\ell$ ?

Your answer

A line described as having a multiple of another line's slope takes two passes: find the first slope from its points, then apply the multiplier. Use the new slope and a point on the new line in $y=mx+b$ ; a Desmos regression can find $b$ . A point's $y$ -coordinate is the intercept only if its $x$ -coordinate is $0$ .

Hints

Hint 1
A slope compares the change in $y$ to the change in $x$ . Subtract the points' coordinates in the same order on top and bottom. As $x$ increases between these points, does $y$ rise or fall?
Hint 2
The two points describe the reference line, not line $\ell$ . Find that line's slope first, then multiply it by $2$ before using the point on $\ell$ .
Hint 3
In slope-intercept form $y=mx+b$ , $b$ is the value of $y$ at $x=0$ . Put the point on $\ell$ into this equation with its doubled slope. Which value is still unknown?

Step-by-step

Find the new slope, then its intercept
Step 1
Find the slope through the two points
The slope is the change in $y$ divided by the change in $x$ . Take the second point minus the first in both places: $m=\frac{-2-2}{5-1}$ . Type $\frac{-2-2}{5-1}$ in Desmos. It prints $-1$ , the slope of the line through the two points. Keeping the same order on top and bottom preserves the sign.
Step 2
Double the slope for line $\ell$
Line $\ell$ has twice the slope you found, so multiply by $2$ : $m_{\ell}=2(-1)=-2$ . Don't use $-1$ as line $\ell$ 's slope; it belongs to the line through the two given points.
Step 3
Use the point to write an intercept equation
In slope-intercept form $y=mx+b$ , $b$ is the $y$ -value when $x=0$ . Line $\ell$ passes through $(4,-3)$ , so substitute $x=4$ , $y=-3$ , and $m=-2$ : $-3=-2(4)+b$ . The $-3$ isn't the intercept because it occurs at $x=4$ , not $x=0$ .
Step 4
Solve for the intercept
Type $-3\sim-2(4)+b$ in Desmos. The $\sim$ asks Desmos to find the unknown; under PARAMETERS, it shows $b=5$ . So line $\ell$ crosses the $y$ -axis at $(0,5)$ , and its $y$ -intercept is $5$ . Grid in 5.

Question 262·Medium·Algebra

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this