Question

The table of ordered pairs represent some points on the graphs of lines q and v.

Line q Line v
x y x y
−9 0 -4 10
−3 18 0 8
2 33 10 3


Which system of equations is represented by lines q and v?
A.
21x−y=9
5x+6y=40

B.
3x−y=−27
x+2y=16

C.
21x−y=9
5x+6y=20

D.
9x−y=−27
x+2y=8

Answer 1

To find the correct system of equations represented by the lines q and v, we will first derive the equations for both lines using the points provided.

**Line q Points:**
1. (-9, 0)
2. (-3, 18)
3. (2, 33)

To find the equation of line q, we can use two points to calculate the slope (m) and then the y-intercept (b).

Let's use points (-9, 0) and (-3, 18):

1. The slope \( m \) is calculated as:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{18 - 0}{-3 - (-9)} = \frac{18}{6} = 3
\]

2. Now, we can use the slope-intercept form of the line \( y = mx + b \). Using point (-9, 0):
\[
0 = 3(-9) + b \implies 0 = -27 + b \implies b = 27
\]

So, the equation for line q is:
\[
y = 3x + 27 \implies 3x - y + 27 = 0 \implies 3x - y = -27
\]

**Line v Points:**
1. (-4, 10)
2. (0, 8)
3. (10, 3)

Using points (-4, 10) and (0, 8) to find the slope (m):

1. The slope \( m \) is:
\[
m = \frac{8 - 10}{0 - (-4)} = \frac{-2}{4} = -\frac{1}{2}
\]

2. Now using the slope-intercept form with point (0, 8):
\[
8 = -\frac{1}{2}(0) + b \implies b = 8
\]

So, the equation for line v is:
\[
y = -\frac{1}{2}x + 8 \implies 2y + x - 16 = 0 \implies x + 2y = 16
\]

**Summary of Equations:**
- Line q: \( 3x - y = -27 \)
- Line v: \( x + 2y = 16 \)

Now, let's compare with the answer choices:

A.
- \( 21x - y = 9 \)
- \( 5x + 6y = 40 \)

B.
- \( 3x - y = -27 \) **(This matches line q)**
- \( x + 2y = 16 \) **(This matches line v)**

C.
- \( 21x - y = 9 \)
- \( 5x + 6y = 20 \)

D.
- \( 9x - y = -27 \)
- \( x + 2y = 8 \)

The correct system of equations represented by lines q and v is:
**B.**
\[
3x - y = -27\\
x + 2y = 16
\]