Certainly! Let's find the slope of the line represented by the given values [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the table.
Given table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
10 & 14 \\
\hline
2 & -2 \\
\hline
3 & 0 \\
\hline
7 & 8 \\
\hline
20 & 34 \\
\hline
\end{array}
\][/tex]
To calculate the slope [tex]\( m \)[/tex], we use the following formula:
[tex]\[
m = \frac{N \sum{xy} - (\sum{x})(\sum{y})}{N \sum{x^2} - (\sum{x})^2}
\][/tex]
We need to calculate the following components:
1. [tex]\( N \)[/tex] (number of data points)
2. [tex]\( \sum{x} \)[/tex]
3. [tex]\( \sum{y} \)[/tex]
4. [tex]\( \sum{xy} \)[/tex]
5. [tex]\( \sum{x^2} \)[/tex]
First, identify the number of data points [tex]\( N \)[/tex]:
[tex]\[
N = 5
\][/tex]
Next, sum the [tex]\( x \)[/tex] values:
[tex]\[
\sum{x} = 10 + 2 + 3 + 7 + 20 = 42
\][/tex]
Sum the [tex]\( y \)[/tex] values:
[tex]\[
\sum{y} = 14 + (-2) + 0 + 8 + 34 = 54
\][/tex]
Calculate [tex]\( \sum{xy} \)[/tex] (multiply each [tex]\( x \)[/tex] value by its corresponding [tex]\( y \)[/tex] value and sum them):
[tex]\[
\sum{xy} = (10 \cdot 14) + (2 \cdot -2) + (3 \cdot 0) + (7 \cdot 8) + (20 \cdot 34) = 140 + (-4) + 0 + 56 + 680 = 872
\][/tex]
Calculate [tex]\( \sum{x^2} \)[/tex] (square each [tex]\( x \)[/tex] value and sum them):
[tex]\[
\sum{x^2} = 10^2 + 2^2 + 3^2 + 7^2 + 20^2 = 100 + 4 + 9 + 49 + 400 = 562
\][/tex]
Now substitute these values into the slope formula:
[tex]\[
m = \frac{N \sum{xy} - (\sum{x})(\sum{y})}{N \sum{x^2} - (\sum{x})^2}
\][/tex]
Using the computed sums:
[tex]\[
m = \frac{5 \cdot 872 - 42 \cdot 54}{5 \cdot 562 - 42^2}
\][/tex]
Calculate the numerator and the denominator:
[tex]\[
\text{Numerator: } 5 \cdot 872 - 42 \cdot 54 = 4360 - 2268 = 2092
\][/tex]
[tex]\[
\text{Denominator: } 5 \cdot 562 - 42^2 = 2810 - 1764 = 1046
\][/tex]
Finally, the slope [tex]\( m \)[/tex]:
[tex]\[
m = \frac{2092}{1046} = 2.0
\][/tex]
Thus, the slope of the line is:
[tex]\[
\boxed{2}
\][/tex]
