Question

Solving the Created System
On a coordinate plane, a line goes through (negative 3, 0) and (0, negative 6).

The graph for the first equation of a system is shown in orange. The second equation is:

3y+30=6x

What is the solution to the system?
(2, –10)
(1, –8)
(–2, –14)
(2, –6)

Answer 1

To solve the system of equations and find the solution, we first need to understand both of the equations provided.

1. We know that the first line goes through the points (-3, 0) and (0, -6). We can find the equation of that line using the slope-intercept form (y = mx + b).

The slope \( m \) between the two points can be calculated as:

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-6 - 0}{0 - (-3)} = \frac{-6}{3} = -2
\]

Now we have the slope \( m = -2 \) and we can use one of the points to find the y-intercept \( b \). Using the point (0, -6):

\[
y = mx + b \implies -6 = -2(0) + b \implies b = -6
\]

Therefore, the equation of the first line is:

\[
y = -2x - 6
\]

2. The second equation is given as:

\[
3y + 30 = 6x
\]

We can rearrange this equation into slope-intercept form. First, isolate \( 3y \):

\[
3y = 6x - 30
\]

Now, divide everything by 3:

\[
y = 2x - 10
\]

Now, we have the two equations of the system:

1. \( y = -2x - 6 \)
2. \( y = 2x - 10 \)

To find the solution to the system, we set the two equations equal to each other:

\[
-2x - 6 = 2x - 10
\]

Now, we solve for \( x \):

1. Add \( 2x \) to both sides:

\[
-6 = 4x - 10
\]

2. Add \( 10 \) to both sides:

\[
4 = 4x
\]

3. Divide by \( 4 \):

\[
x = 1
\]

Now that we have \( x = 1 \), we substitute it back into one of the original equations to find \( y \). We can use either equation; we'll use \( y = 2x - 10 \):

\[
y = 2(1) - 10 = 2 - 10 = -8
\]

Thus, the solution to the system is:

\[
(x, y) = (1, -8)
\]

Among the provided options, the solution is:

\((1, -8)\)