To complete the table and graph the equation [tex]\( 2x + 3y = 12 \)[/tex], we need to find the corresponding values of [tex]\( y \)[/tex] for given values of [tex]\( x \)[/tex], and vice versa.
Given equation:
[tex]\[ 2x + 3y = 12 \][/tex]
### Step-by-Step Solution:
1. Given: [tex]\( x = -3 \)[/tex]
[tex]\[
2(-3) + 3y = 12 \\
-6 + 3y = 12 \\
3y = 18 \\
y = 6
\][/tex]
So, when [tex]\( x = -3 \)[/tex], [tex]\( y = 6 \)[/tex].
2. Find [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex]
[tex]\[
2(0) + 3y = 12 \\
3y = 12 \\
y = 4
\][/tex]
So, when [tex]\( x = 0 \)[/tex], [tex]\( y = 4 \)[/tex].
3. Find [tex]\( x \)[/tex] when [tex]\( y = 0 \)[/tex]
[tex]\[
2x + 3(0) = 12 \\
2x = 12 \\
x = 6
\][/tex]
So, when [tex]\( y = 0 \)[/tex], [tex]\( x = 6 \)[/tex].
4. Find [tex]\( y \)[/tex] when [tex]\( x = 3 \)[/tex]
[tex]\[
2(3) + 3y = 12 \\
6 + 3y = 12 \\
3y = 6 \\
y = 2
\][/tex]
So, when [tex]\( x = 3 \)[/tex], [tex]\( y = 2 \)[/tex].
### Completed Table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-3 & 6 \\
\hline
0 & 4 \\
\hline
6 & 0 \\
\hline
3 & 2 \\
\hline
\end{array}
\][/tex]
So, the completed table is:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-3 & 6 \\
\hline
0 & 4 \\
\hline
6 & 0 \\
\hline
3 & 2 \\
\hline
\end{array}
\][/tex]
With these points, you can graph the equation [tex]\( 2x + 3y = 12 \)[/tex]. Each pair [tex]\((x, y)\)[/tex] represents a point on the graph of the line. Make sure you plot each of these points accurately and draw the line passing through them.
