To determine which equation corresponds to [tex]\( Y_2 \)[/tex] in the table, we need to analyze the relationship between [tex]\( X \)[/tex] and [tex]\( Y_2 \)[/tex]. Let's break down each step:
1. Examine the given values for [tex]\( X \)[/tex] and [tex]\( Y_2 \)[/tex]:
[tex]\[
\begin{array}{|r|c|}
\hline
X & Y_2 \\
\hline
-3 & -3 \\
-2 & -4 \\
-1 & -5 \\
0 & -6 \\
1 & -7 \\
2 & -8 \\
3 & -9 \\
\hline
\end{array}
\][/tex]
2. Identify the pattern and relationship:
- Observe how [tex]\( Y_2 \)[/tex] changes as [tex]\( X \)[/tex] increases.
- [tex]\( Y_2 \)[/tex] decreases by 1 for each increase of 1 in [tex]\( X \)[/tex].
3. Determine the type of relationship:
- The change in [tex]\( Y_2 \)[/tex] is consistent and linear.
- This suggests a linear equation of the form [tex]\( Y_2 = mX + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
4. Calculating the slope ([tex]\( m \)[/tex]):
- The slope [tex]\( m \)[/tex] represents the change in [tex]\( Y_2 \)[/tex] per change in [tex]\( X \)[/tex].
- From the table, we see that [tex]\( Y_2 \)[/tex] decreases by 1 unit for each 1 unit increase in [tex]\( X \)[/tex], so [tex]\( m = -1 \)[/tex].
5. Calculating the y-intercept ([tex]\( b \)[/tex]):
- To find [tex]\( b \)[/tex], we can use one of the points from the table. Let's take the point [tex]\((0, -6)\)[/tex].
- Substituting [tex]\( X = 0 \)[/tex] and [tex]\( Y_2 = -6 \)[/tex] into the equation [tex]\( Y_2 = mX + b \)[/tex] gives us:
[tex]\[
-6 = -1 \cdot 0 + b \implies b = -6
\][/tex]
6. Formulate the equation:
- With [tex]\( m = -1 \)[/tex] and [tex]\( b = -6 \)[/tex], the equation that corresponds to [tex]\( Y_2 \)[/tex] is:
[tex]\[
Y_2 = -X - 6
\][/tex]
7. Verify the equation:
- Substitute values of [tex]\( X \)[/tex] from the table into the equation [tex]\( Y_2 = -X - 6 \)[/tex] to verify if it matches the [tex]\( Y_2 \)[/tex] values.
[tex]\[
\begin{align*}
\text{For } X = -3, & \quad Y_2 = -(-3) - 6 = 3 - 6 = -3 \\
\text{For } X = -2, & \quad Y_2 = -(-2) - 6 = 2 - 6 = -4 \\
\text{For } X = -1, & \quad Y_2 = -(-1) - 6 = 1 - 6 = -5 \\
\text{For } X = 0, & \quad Y_2 = -0 - 6 = -6 \\
\text{For } X = 1, & \quad Y_2 = -1 - 6 = -7 \\
\text{For } X = 2, & \quad Y_2 = -2 - 6 = -8 \\
\text{For } X = 3, & \quad Y_2 = -3 - 6 = -9 \\
\end{align*}
\][/tex]
- The values obtained from the equation match the given [tex]\( Y_2 \)[/tex] values in the table.
The equation corresponding to [tex]\( Y_2 \)[/tex] in the given table is:
[tex]\[
Y_2 = -X - 6
\][/tex]
