Question 683·Medium·Algebra

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In the $xy$ -plane, line $k$ passes through $(2,-1)$ and is parallel to the line with equation $4x - 3y = 12$ . What is the $y$ -intercept of line $k$ ?

In a parallel lines question, one line gives you the direction and a point tells you where the new line sits. Graph the given equation in Desmos and read two axis crossings to find its slope. Then keep the slope, not the intercept, and use the new point to find where its line crosses the $y$ -axis.

Hints

Hint 1
A slope is the change in $y$ divided by the change in $x$ . Click the given line’s two axis crossings in Desmos. How far do you move right and up between them?
Hint 2
Parallel lines have the same slope, but they can cross the $y$ -axis at different places. Write line $k$ as $y=mx+b$ , using the slope you found and leaving its intercept $b$ unknown.
Hint 3
A point on a line makes its equation true. Put the coordinates $(2,-1)$ into your equation for line $k$ , then solve for $b$ , the value when $x=0$ .

Step-by-step

Find the slope, then the new intercept
Step 1
Read two points on the given line
Type $4x-3y=12$ in Desmos. Click the line, then its two axis crossings. Desmos labels them $(0,-4)$ and $(3,0)$ . These points belong to the given line; $-4$ is not automatically line $k$ ’s intercept.
Step 2
Find the given line’s slope
A slope is the change in $y$ divided by the change in $x$ . From $(0,-4)$ to $(3,0)$ , calculate rise over run:
$m=\frac{0-(-4)}{3-0}$
Type $\frac{0-(-4)}{3-0}$ in Desmos and press the fraction button. Desmos shows $\frac{4}{3}$ , so:
$m=\frac{4}{3}.$
Step 3
Give line $k$ the same slope
Parallel lines have the same slope, so line $k$ also has slope $\frac43$ . In slope-intercept form, $b$ is the value of $y$ when $x=0$ :
$y=\frac43x+b.$
Step 4
Use the point on line $k$
The point $(2,-1)$ means $y=-1$ when $x=2$ . Substitute both coordinates into line $k$ ’s equation:
$-1=\frac43(2)+b.$
Step 5
Read the exact value
Type $-1\sim\frac43(2)+b$ . The $\sim$ tells Desmos to find the unknown $b$ that makes the two sides equal. Under PARAMETERS, Desmos shows $b\approx-3.6667$ . Type $b$ on its own line and press the fraction button. Desmos shows $-\frac{11}{3}$ . Since $b$ is the value when $x=0$ , the $y$ -intercept of line $k$ is $-\frac{11}{3}$ .

Choice C.

Question 683·Medium·Algebra

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this