To solve the problem, we need to complete the table of values for the equation [tex]\( x + 2y = 7 \)[/tex] and then draw the graph for [tex]\( x \)[/tex] between [tex]\(-2\)[/tex] and [tex]\(3\)[/tex].
### Part (a): Completing the Table of Values
The given equation is [tex]\( x + 2y = 7 \)[/tex]. We will use this equation to find the value of [tex]\( y \)[/tex] for each given [tex]\( x \)[/tex].
Let's find the corresponding [tex]\( y \)[/tex] values for the given [tex]\( x \)[/tex] values.
1. When [tex]\( x = -2 \)[/tex]:
[tex]\[ -2 + 2y = 7 \][/tex]
[tex]\[ 2y = 7 + 2 \][/tex]
[tex]\[ 2y = 9 \][/tex]
[tex]\[ y = \frac{9}{2} \][/tex]
[tex]\[ y = 4.5 \][/tex]
2. When [tex]\( x = -1 \)[/tex]:
[tex]\[ -1 + 2y = 7 \][/tex]
[tex]\[ 2y = 7 + 1 \][/tex]
[tex]\[ 2y = 8 \][/tex]
[tex]\[ y = \frac{8}{2} \][/tex]
[tex]\[ y = 4 \][/tex]
3. When [tex]\( x = 0 \)[/tex]:
[tex]\[ 0 + 2y = 7 \][/tex]
[tex]\[ 2y = 7 \][/tex]
[tex]\[ y = \frac{7}{2} \][/tex]
[tex]\[ y = 3.5 \][/tex]
4. When [tex]\( x = 1 \)[/tex]:
[tex]\[ 1 + 2y = 7 \][/tex]
[tex]\[ 2y = 7 - 1 \][/tex]
[tex]\[ 2y = 6 \][/tex]
[tex]\[ y = \frac{6}{2} \][/tex]
[tex]\[ y = 3 \][/tex]
5. When [tex]\( x = 2 \)[/tex]:
[tex]\[ 2 + 2y = 7 \][/tex]
[tex]\[ 2y = 7 - 2 \][/tex]
[tex]\[ 2y = 5 \][/tex]
[tex]\[ y = \frac{5}{2} \][/tex]
[tex]\[ y = 2.5 \][/tex]
6. When [tex]\( x = 3 \)[/tex]:
[tex]\[ 3 + 2y = 7 \][/tex]
[tex]\[ 2y = 7 - 3 \][/tex]
[tex]\[ 2y = 4 \][/tex]
[tex]\[ y = \frac{4}{2} \][/tex]
[tex]\[ y = 2 \][/tex]
Now we can fill in the table with the values we just calculated:
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
[tex]$x$[/tex] & -2 & -1 & 0 & 1 & 2 & 3 \\
\hline
[tex]$y$[/tex] & 4.5 & 4.0 & 3.5 & 3.0 & 2.5 & 2.0 \\
\hline
\end{tabular}
### Part (b): Drawing the Graph
To draw the graph of the equation [tex]\( x + 2y = 7 \)[/tex] for values of [tex]\( x \)[/tex] between [tex]\(-2\)[/tex] and [tex]\(3\)[/tex], we plot the points calculated in the table:
- Point (-2, 4.5)
- Point (-1, 4.0)
- Point (0, 3.5)
- Point (1, 3.0)
- Point (2, 2.5)
- Point (3, 2.0)
### Steps to Draw the Graph:
1. Label the x-axis and y-axis on the grid.
2. Plot the points (-2, 4.5), (-1, 4.0), (0, 3.5), (1, 3.0), (2, 2.5), and (3, 2.0) on the grid.
3. Connect the points with a straight line since they all satisfy the linear equation [tex]\( x + 2y = 7 \)[/tex].
By plotting these points and drawing the line through them, you will have the graph of the equation [tex]\( x + 2y = 7 \)[/tex] for values of [tex]\( x \)[/tex] between [tex]\(-2\)[/tex] and [tex]\(3\)[/tex]. This line should demonstrate a linear relationship with a negative slope.
