To solve the given equation [tex]\(3x - 2y = -6\)[/tex] and complete the table, we'll determine the values of [tex]\(y\)[/tex] at specific [tex]\(x\)[/tex] values. Here are the steps we'll follow:
1. Rearrange the equation to solve for [tex]\(y\)[/tex]:
[tex]\[3x - 2y = -6 \quad \Rightarrow \quad -2y = -3x - 6 \quad \Rightarrow \quad y = \frac{3x + 6}{2}\][/tex]
2. Substitute the [tex]\(x\)[/tex] values into the equation and calculate the corresponding [tex]\(y\)[/tex] values.
Let's fill in the table for [tex]\(x\)[/tex] values from [tex]\(-2\)[/tex] to [tex]\(2\)[/tex], including [tex]\(x = 0\)[/tex]:
1. When [tex]\(x = -2\)[/tex]:
[tex]\[y = \frac{3(-2) + 6}{2} = \frac{-6 + 6}{2} = \frac{0}{2} = 0\][/tex]
2. When [tex]\(x = -1\)[/tex]:
[tex]\[y = \frac{3(-1) + 6}{2} = \frac{-3 + 6}{2} = \frac{3}{2} = 1.5\][/tex]
3. When [tex]\(x = 0\)[/tex]:
[tex]\[y = \frac{3(0) + 6}{2} = \frac{0 + 6}{2} = \frac{6}{2} = 3\][/tex]
4. When [tex]\(x = 1\)[/tex]:
[tex]\[y = \frac{3(1) + 6}{2} = \frac{3 + 6}{2} = \frac{9}{2} = 4.5\][/tex]
5. When [tex]\(x = 2\)[/tex]:
[tex]\[y = \frac{3(2) + 6}{2} = \frac{6 + 6}{2} = \frac{12}{2} = 6\][/tex]
Now we can complete the table:
[tex]\[
\begin{tabular}{|c|c|}
\hline$x$ & $y$ \\
\hline -2 & 0.0 \\
\hline -1 & 1.5 \\
\hline 0 & 3.0 \\
\hline 1 & 4.5 \\
\hline 2 & 6.0 \\
\hline
\end{tabular}
\][/tex]
This completes the table for the given equation [tex]\(3x - 2y = -6\)[/tex].
