Let's analyze the given table and identify the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
[tex]\[
\begin{array}{c|c}
x & y \\
\hline
0 & 3 \\
1 & 5 \\
2 & 7 \\
3 & 9 \\
4 & 11 \\
5 & 13 \\
\end{array}
\][/tex]
First, observe the values of [tex]\( y \)[/tex] as [tex]\( x \)[/tex] increases. We notice that:
For [tex]\( x = 0 \)[/tex], [tex]\( y = 3 \)[/tex]
For [tex]\( x = 1 \)[/tex], [tex]\( y = 5 \)[/tex]
For [tex]\( x = 2 \)[/tex], [tex]\( y = 7 \)[/tex]
For [tex]\( x = 3 \)[/tex], [tex]\( y = 9 \)[/tex]
For [tex]\( x = 4 \)[/tex], [tex]\( y = 11 \)[/tex]
For [tex]\( x = 5 \)[/tex], [tex]\( y = 13 \)[/tex]
We can observe that [tex]\( y \)[/tex] increases by 2 for each increase of 1 in [tex]\( x \)[/tex]. Therefore, the pattern suggests that [tex]\( y \)[/tex] is a linear function of [tex]\( x \)[/tex].
To find the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex], let's assume the equation is:
[tex]\[ y = mx + b \][/tex]
Where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
From the table, we see that as [tex]\( x \)[/tex] increases by 1, [tex]\( y \)[/tex] increases by 2. So, the slope [tex]\( m = 2 \)[/tex].
Next, to determine the y-intercept [tex]\( b \)[/tex], we can use one of the points from the table. Let's use the point [tex]\( (0, 3) \)[/tex]:
[tex]\[ y = 2x + b \][/tex]
When [tex]\( x = 0 \)[/tex], [tex]\( y = 3 \)[/tex]:
[tex]\[ 3 = 2(0) + b \][/tex]
[tex]\[ b = 3 \][/tex]
Thus, the equation describing the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is:
[tex]\[ y = 2x + 3 \][/tex]
Now, we need to find the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 2(0) + 3 \][/tex]
[tex]\[ y = 3 \][/tex]
So, the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex] is [tex]\( \boxed{3} \)[/tex].
