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:
- (-9, 0)
- (-3, 18)
- (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):
-
The slope \( m \) is calculated as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{18 - 0}{-3 - (-9)} = \frac{18}{6} = 3 \]
-
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:
- (-4, 10)
- (0, 8)
- (10, 3)
Using points (-4, 10) and (0, 8) to find the slope (m):
-
The slope \( m \) is: \[ m = \frac{8 - 10}{0 - (-4)} = \frac{-2}{4} = -\frac{1}{2} \]
-
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. \[ 3x - y = -27\ x + 2y = 16 \]