To find the equation that describes the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] from the given table, we follow a systematic approach.
Given data:
[tex]\[
\begin{array}{c|c}
x & y \\
\hline
0 & 3 \\
1 & 5 \\
2 & 7 \\
3 & 9 \\
4 & 11 \\
5 & 13 \\
\end{array}
\][/tex]
1. Determine the rate of change (slope):
Notice how [tex]\( y \)[/tex] changes as [tex]\( x \)[/tex] increases:
[tex]\[
\begin{aligned}
5 - 3 &= 2, \\
7 - 5 &= 2, \\
9 - 7 &= 2, \\
11 - 9 &= 2, \\
13 - 11 &= 2.
\end{aligned}
\][/tex]
Each time [tex]\( x \)[/tex] increases by 1, [tex]\( y \)[/tex] increases by 2. Therefore, the slope [tex]\( m \)[/tex] of the line is 2.
2. Identify the y-intercept:
When [tex]\( x = 0 \)[/tex], [tex]\( y = 3 \)[/tex].
The y-intercept ([tex]\( b \)[/tex]) is 3.
3. Form the linear equation:
The general form of a linear equation is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- Slope [tex]\( m \)[/tex] = 2
- Y-intercept [tex]\( b \)[/tex] = 3
Substituting these values into the equation gives:
[tex]\[
y = 2x + 3
\][/tex]
Therefore, the equation that describes the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is:
[tex]\[
y = 2x + 3
\][/tex]
