Let's complete the equation that describes the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] based on the given data.
We have the following data points:
[tex]\[
\begin{array}{c|c}
x & y \\
\hline
0 & 3 \\
1 & 5 \\
2 & 7 \\
3 & 9 \\
4 & 11 \\
5 & 13 \\
\end{array}
\][/tex]
We observe the following pattern:
1. When [tex]\( x \)[/tex] changes by 1, [tex]\( y \)[/tex] changes by 2.
2. When [tex]\( x \)[/tex] is 0, [tex]\( y \)[/tex] is 3.
### Step-by-Step Solution
Step 1: Find the slope (rate of change) of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex]:
The slope [tex]\( m \)[/tex] is the change in [tex]\( y \)[/tex] divided by the change in [tex]\( x \)[/tex].
[tex]\[
\text{slope} = m = \frac{\Delta y}{\Delta x}
\][/tex]
Given that when [tex]\( x \)[/tex] changes by 1, [tex]\( y \)[/tex] changes by 2:
[tex]\[
m = \frac{2}{1} = 2
\][/tex]
So, the slope [tex]\( m \)[/tex] is 2.
Step 2: Find the y-intercept:
The y-intercept [tex]\( b \)[/tex] is the value of [tex]\( y \)[/tex] when [tex]\( x \)[/tex] is 0. From the given data point (0, 3):
[tex]\[
b = 3
\][/tex]
Step 3: Write the equation:
Using the slope-intercept form of a linear equation [tex]\( y = mx + b \)[/tex], we can substitute the values of [tex]\( m \)[/tex] and [tex]\( b \)[/tex] we found:
[tex]\[
y = 2x + 3
\][/tex]
### Conclusion
The complete equation describing the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is:
[tex]\[
y = 2x + 3
\][/tex]
This equation tells us that for each unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 2, and when [tex]\( x \)[/tex] is 0, [tex]\( y \)[/tex] is 3.
