Question

What system of equations is shown on the graph below?


On a coordinate plane, a line goes through points (negative 2, 0) and (0, negative 4) and another line goes through (0, negative 2) and (4, 0).

x minus 2 y = negative 4 and 2 x + y = 4
x minus 2 y = 4 and 2 x + y = negative 4
x minus 2 y = negative 2 and 2 x + y = negative 4
x minus 2 y = 2 and 2 x + y = 4

Answer 1

To determine the system of equations represented by the lines on the graph, we can find the equations of each line using the points provided.

**First line**: You stated that it goes through points (-2, 0) and (0, -4).

To find the slope:
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - 0}{0 - (-2)} = \frac{-4}{2} = -2
\]

Using the point-slope form \(y - y_1 = m(x - x_1)\):
\[
y - 0 = -2(x + 2)
\]
Distributing:
\[
y = -2x - 4
\]
Rearranging gives:
\[
2x + y = -4 \quad \text{(first equation)}
\]

**Second line**: It goes through points (0, -2) and (4, 0).

Finding the slope:
\[
\text{slope} = \frac{0 - (-2)}{4 - 0} = \frac{2}{4} = \frac{1}{2}
\]

Using the point-slope form:
\[
y - (-2) = \frac{1}{2}(x - 0)
\]
This simplifies to:
\[
y + 2 = \frac{1}{2}x
\]
Rearranging gives:
\[
-\frac{1}{2}x + y = -2
\]
Multiplying through by -2 to eliminate the fraction gives:
\[
x - 2y = 4 \quad \text{(second equation)}
\]

Now, we have the two equations:
1. \(2x + y = -4\)
2. \(x - 2y = 4\)

We can match these with the options presented:
- The correct system of equations is:
\[
x - 2y = 4 \quad \text{and} \quad 2x + y = -4
\]

Thus, the correct answer is:
**x - 2y = 4 and 2x + y = -4**.