To complete the table based on the given domain and function [tex]\( y = -\frac{2}{3} x + 7 \)[/tex], we have the following results:
1. For [tex]\( x = -6 \)[/tex]:
[tex]\[ y = -\frac{2}{3}(-6) + 7 = 4 + 7 = 11 \][/tex]
2. For [tex]\( y = 5 \)[/tex]:
Solving the equation for [tex]\( x \)[/tex]:
[tex]\[ 5 = -\frac{2}{3} x + 7 \][/tex]
[tex]\[ -\frac{2}{3} x = 5 - 7 \][/tex]
[tex]\[ -\frac{2}{3} x = -2 \][/tex]
[tex]\[ x = -2 \cdot \left( \frac{3}{-2} \right) = 3 \][/tex]
3. For [tex]\( x = 15 \)[/tex]:
[tex]\[ y = -\frac{2}{3}(15) + 7 = -10 + 7 = -3 \][/tex]
4. For [tex]\( y = 15 \)[/tex]:
Solving the equation for [tex]\( x \)[/tex]:
[tex]\[ 15 = -\frac{2}{3} x + 7 \][/tex]
[tex]\[ -\frac{2}{3} x = 15 - 7 \][/tex]
[tex]\[ -\frac{2}{3} x = 8 \][/tex]
[tex]\[ x = 8 \cdot \left( \frac{3}{-2} \right) = -12 \][/tex]
So, the completed table is:
[tex]\[
\begin{tabular}{|c|c|}
\hline$x$ & $y$ \\
\hline-6 & 11 \\
\hline3 & 5 \\
\hline 15 & -3 \\
\hline-12 & 15 \\
\hline
\end{tabular}
\][/tex]
