Question

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 find the solution to the given system of equations, we need to first find the equation of the line that passes through the points (-3, 0) and (0, -6), and then solve the second equation \(3y + 30 = 6x\).

**Step 1: Find the equation of the first line.**

We can start by calculating the slope (m) of the line that passes through the points (-3, 0) and (0, -6):

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

Now, using the slope-intercept form of a line \(y = mx + b\), we can substitute one of the points to find b. We can use the point (0, -6):

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

So the equation of the first line is:

\[
y = -2x - 6
\]

**Step 2: Simplify the second equation.**

We start with the second equation:

\[
3y + 30 = 6x
\]

Let's solve for y. First, we can isolate \(3y\):

\[
3y = 6x - 30
\]

Now, divide by 3:

\[
y = 2x - 10
\]

**Step 3: Find the intersection of the two lines.**

We now have the two equations:

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

To find their intersection, we set the two equations equal to each other:

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

Now we can combine like terms:

\[
-2x - 2x = -10 + 6
\]
\[
-4x = -4 \implies x = 1
\]

**Step 4: Substitute x back to find y.**

We can substitute \(x = 1\) into either equation to find \(y\). Using the second equation:

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

Thus, the solution to the system of equations is:

\[
(1, -8)
\]

Among the provided options, this corresponds to:

\[
\boxed{(1, -8)}
\]