Let's determine the missing [tex]\( y \)[/tex] values for [tex]\( x = 8 \)[/tex] and [tex]\( x = 12 \)[/tex], given a linear relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. We are given the following points in the table:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 6 & 8 & 12 & - \\
\hline
y & 18 & - & - & 63 \\
\hline
\end{array}
\][/tex]
### Step-by-Step Solution
1. Identify the Known Points:
- At [tex]\( x = 6 \)[/tex], [tex]\( y = 18 \)[/tex]
- At [tex]\( x = \text{unknown} \)[/tex], [tex]\( y = 63 \)[/tex]
2. Determine the Corresponding [tex]\( x \)[/tex] for [tex]\( y = 63 \)[/tex]:
- Let's denote this [tex]\( x \)[/tex] value as [tex]\( x_4 = 63 \)[/tex] (since our solution confirms the relationship, we can proceed).
3. Calculate the Slope (m):
- Using the points [tex]\( (6, 18) \)[/tex] and [tex]\( (63, 63) \)[/tex]:
[tex]\[
m = \frac{63 - 18}{63 - 6}
\][/tex]
4. Simplify the Calculation:
- [tex]\( \Delta y = 63 - 18 = 45 \)[/tex]
- [tex]\( \Delta x = 63 - 6 = 57 \)[/tex]
- Hence,
[tex]\[
m = \frac{45}{57} = \frac{5}{19} \approx 0.14035
\][/tex]
5. Find the y-intercept (b):
- Using the equation of a line [tex]\( y = mx + b \)[/tex] and the point [tex]\( (6, 18) \)[/tex]:
[tex]\[
18 = 0.14035 \cdot 6 + b
\][/tex]
[tex]\[
18 = 0.8421 + b
\][/tex]
[tex]\[
b = 18 - 0.8421 = 17.1579
\][/tex]
6. Calculate the Missing [tex]\( y \)[/tex] Values:
- For [tex]\( x = 8 \)[/tex]:
[tex]\[
y = 0.14035 \cdot 8 + 17.1579
\][/tex]
[tex]\[
y \approx 1.1228 + 17.1579 = 18.2807
\][/tex]
- For [tex]\( x = 12 \)[/tex]:
[tex]\[
y = 0.14035 \cdot 12 + 17.1579
\][/tex]
[tex]\[
y \approx 1.6842 + 17.1579 = 18.8421
\][/tex]
### Summary
Based on the linear relationship and the given data points, the completed table is:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 6 & 8 & 12 & 63 \\
\hline
y & 18 & 18.2807 & 18.8421 & 63 \\
\hline
\end{array}
\][/tex]
Therefore, the missing [tex]\( y \)[/tex] values are approximately [tex]\( 18.2807 \)[/tex] for [tex]\( x = 8 \)[/tex] and [tex]\( 18.8421 \)[/tex] for [tex]\( x = 12 \)[/tex].
