To solve the system of equations by graphing, we need to find the intersection point of the lines represented by the two equations:
[tex]\[
\begin{array}{l}
1. \quad 30 + 6y = 16 \\
2. \quad 6x - 4y = 0
\end{array}
\][/tex]
Let's simplify each equation step by step.
### Simplify Equation 1:
[tex]\[ 30 + 6y = 16 \][/tex]
Subtract 30 from both sides:
[tex]\[ 6y = 16 - 30 \][/tex]
[tex]\[ 6y = -14 \][/tex]
Divide both sides by 6:
[tex]\[ y = \frac{-14}{6} = \frac{-7}{3} \][/tex]
### Simplify Equation 2:
[tex]\[ 6x - 4y = 0 \][/tex]
Add [tex]\(4y\)[/tex] to both sides:
[tex]\[ 6x = 4y \][/tex]
Divide both sides by 6:
[tex]\[ x = \frac{4}{6} y \][/tex]
[tex]\[ x = \frac{2}{3} y \][/tex]
### Substitute [tex]\( y = \frac{-7}{3} \)[/tex] into [tex]\( x = \frac{2}{3} y \)[/tex]:
[tex]\[ x = \frac{2}{3} \left(\frac{-7}{3}\right) \][/tex]
[tex]\[ x = \frac{2 \cdot -7}{3 \cdot 3} \][/tex]
[tex]\[ x = \frac{-14}{9} \][/tex]
Thus, the solution to the system of equations is:
[tex]\[ x = \frac{-14}{9}, \; y = \frac{-7}{3} \][/tex]
Now, let's match this solution with the given answer choices:
A. [tex]\(\left(\frac{1}{3}, -\frac{1}{2}\right)\)[/tex] - This does not match the solution.
B. [tex]\(\left(\frac{4}{3}, \frac{1}{1}\right)\)[/tex] - This does not match the solution.
C. [tex]\(\left(-\frac{1}{2}, -\frac{1}{4}\right)\)[/tex] - This does not match the solution.
D. [tex]\(\left(\frac{1}{4}, \frac{4}{3}\right)\)[/tex] - This does not match the solution.
None of the given options match the solution [tex]\(\left(\frac{-14}{9}, \frac{-7}{3}\right)\)[/tex]. Therefore, it appears that the correct solution is not listed among the provided options.
