Let's find the missing values in the table step-by-step.
Given the line equation:
[tex]\[ y = -\frac{2}{3}x + 7 \][/tex]
1. Finding [tex]\(y\)[/tex] when [tex]\(x = -6\)[/tex]:
[tex]\[ y = -\frac{2}{3}(-6) + 7\][/tex]
[tex]\[ y = 4 + 7 \][/tex]
[tex]\[ y = 11 \][/tex]
So, the corresponding [tex]\( y \)[/tex] value for [tex]\( x = -6 \)[/tex] is 11.
2. Finding [tex]\(x\)[/tex] when [tex]\( y = 5 \)[/tex]:
We use the equation and solve for [tex]\( x \)[/tex]:
[tex]\[ 5 = -\frac{2}{3}x + 7 \][/tex]
Subtract 7 from both sides:
[tex]\[ 5 - 7 = -\frac{2}{3}x \][/tex]
[tex]\[ -2 = -\frac{2}{3}x \][/tex]
Divide both sides by [tex]\(-\frac{2}{3}\)[/tex]:
[tex]\[ x = \frac{-2}{-\frac{2}{3}} \][/tex]
[tex]\[ x = 3 \][/tex]
So, the corresponding [tex]\( x \)[/tex] value for [tex]\( y = 5 \)[/tex] is 3.
3. Finding [tex]\(y\)[/tex] when [tex]\(x = 15\)[/tex]:
[tex]\[ y = -\frac{2}{3}(15) + 7 \][/tex]
[tex]\[ y = -10 + 7 \][/tex]
[tex]\[ y = -3 \][/tex]
So, the corresponding [tex]\( y \)[/tex] value for [tex]\( x = 15 \)[/tex] is -3.
4. Finding [tex]\(x\)[/tex] when [tex]\( y = 15 \)[/tex]:
We use the equation and solve for [tex]\( x \)[/tex]:
[tex]\[ 15 = -\frac{2}{3}x + 7 \][/tex]
Subtract 7 from both sides:
[tex]\[ 15 - 7 = -\frac{2}{3}x \][/tex]
[tex]\[ 8 = -\frac{2}{3}x \][/tex]
Divide both sides by [tex]\(-\frac{2}{3}\)[/tex]:
[tex]\[ x = \frac{8}{-\frac{2}{3}} \][/tex]
[tex]\[ x = -12 \][/tex]
So, the corresponding [tex]\( x \)[/tex] value for [tex]\( y = 15 \)[/tex] is -12.
Now we fill in the table with these values:
\begin{tabular}{|c|c|}
\hline [tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline -6 & 11 \\
\hline 3 & 5 \\
\hline 15 & -3 \\
\hline -12 & 15 \\
\hline
\end{tabular}
