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)

1 answer

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:

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