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

1 answer

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. \[ 3x - y = -27\ x + 2y = 16 \]