Mr. Coleman is mapping the boundaries of a zoo on a coordinate grid. The zoo’s headquarters are located at the origin. The equations shown below represent two boundaries of the zoo.

Question

y = −2x – 5
−2x + 4y = 12

The zoo’s entrance is located at the intersection of these two boundaries. Which coordinate grid correctly shows the two boundaries and the zoo’s entrance?
Responses
A AA
B BB
C CC
D D

Answer 1

To find the coordinates of the intersection of the two boundaries, we need to solve the equations simultaneously.

1. The first equation is already in slope-intercept form: \[ y = -2x - 5 \]

2. The second equation can be rewritten as follows: \[ -2x + 4y = 12 \ 4y = 2x + 12 \ y = \frac{1}{2}x + 3 \]

Now, we have the two equations:

1. \( y = -2x - 5 \)
2. \( y = \frac{1}{2}x + 3 \)

To find the intersection, we can set the two equations equal to each other: \[ -2x - 5 = \frac{1}{2}x + 3 \]

Now, let's multiply through by 2 to eliminate the fraction: \[ -4x - 10 = x + 6 \]

Next, we'll collect all terms involving \( x \) on one side and constant terms on the other: \[ -4x - x = 6 + 10 \ -5x = 16 \ x = -\frac{16}{5} \ x = -3.2 \]

Now, substitute \( x = -3.2 \) back into one of the original equations to solve for \( y \). We'll use the first equation: \[ y = -2(-3.2) - 5 \ y = 6.4 - 5 \ y = 1.4 \]

Thus, the coordinates of the zoo’s entrance, which is the intersection of the two boundaries, are: \[ (-3.2, 1.4) \]

Now, you need to check which of the response options A, B, C, or D correctly represents the boundaries and the point (-3.2, 1.4) on the coordinate grid. If you can provide the visual options, I can help you choose which one corresponds with these equations and that intersection point.