Question

A graph displays a straight line. The line has a positive slope and passes through the points approximately (-8, -2) and (8, 6). The x-axis ranges from -10 to 10, and the y-axis ranges from -10 to 10. The graph is on a gridded coordinate plane.

Identify the equation of the graph shown.
Responses
A y = 2x - 3y = 2x - 3
B y = 2x + 6y = 2x + 6
C y = 0.5x + 6y = 0.5x + 6
D y = 0.5x - 3y = 0.5x - 3
E y = 0.5x + 3

Answer 1

To find the equation of the line based on the two points given, we can start by calculating the slope (\(m\)) of the line using the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the points (-8, -2) and (8, 6):

\[
x_1 = -8, \quad y_1 = -2
\]
\[
x_2 = 8, \quad y_2 = 6
\]

Substituting into the slope formula:

\[
m = \frac{6 - (-2)}{8 - (-8)} = \frac{6 + 2}{8 + 8} = \frac{8}{16} = 0.5
\]

Now we know the slope is \(0.5\).

Next, we can use one of the points to find the y-intercept (\(b\)) of the line using the equation of a line in slope-intercept form:

\[
y = mx + b
\]

Substituting one of the points into the equation. We can use (-8, -2):

\[
-2 = 0.5(-8) + b
\]
\[
-2 = -4 + b
\]
\[
b = -2 + 4 = 2
\]

Thus, the equation of the line is:

\[
y = 0.5x + 2
\]

However, this option is not listed in the responses. Let's double-check our calculations, starting from

\[
y = 0.5x + b
\]

Let's use the point (8, 6) to verify:

\[
6 = 0.5(8) + b
\]
\[
6 = 4 + b
\]
\[
b = 6 - 4 = 2
\]

The constant term is consistently computed as 2, and we still conclude that:

\[
y = 0.5x + 2
\]

Revisiting the options:

- A: \(y = 2x - 3\)
- B: \(y = 2x + 6\)
- C: \(y = 0.5x + 6\)
- D: \(y = 0.5x - 3\)
- E: \(y = 0.5x + 3\)

None of the provided options correctly state the discovered equation \(y = 0.5x + 2\).

If forced to select, the closest choice would be \(E: y = 0.5x + 3\), but it's still incorrect based on our calculations. Thus, if you need to indicate the most likely choice according to the manner given in a multiple-choice format, then option **E** is the only with a matching slope but incorrect y-intercept.

None of the options directly match the equation we derived.