To determine the linear equation [tex]\( y = mx + c \)[/tex] for the given table, we need to calculate the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( c \)[/tex]. Let's outline the steps:
1. Identify the given points:
The table provides the following pairs of [tex]\( (x, y) \)[/tex] values:
[tex]\[
\begin{array}{ccc}
x & : & 5, 10, 22, 23 \\
y & : & 57, 67, 91, 93 \\
\end{array}
\][/tex]
2. Calculate the mean of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
\text{Mean of } x (\overline{x}) = 15.0
\][/tex]
[tex]\[
\text{Mean of } y (\overline{y}) = 77.0
\][/tex]
3. Find the sums needed for the slope (m):
The numerator part of the slope formula, which is the sum of the product of the differences of each [tex]\( x \)[/tex] and [tex]\( y \)[/tex] from their respective means, is calculated as follows:
[tex]\[
\text{Numerator} = \sum (x_i - \overline{x})(y_i - \overline{y}) = 476.0
\][/tex]
The denominator part, which is the sum of the squared differences of each [tex]\( x \)[/tex] from the mean of [tex]\( x \)[/tex], is:
[tex]\[
\text{Denominator} = \sum (x_i - \overline{x})^2 = 238.0
\][/tex]
4. Calculate the slope:
[tex]\[
m = \frac{\text{Numerator}}{\text{Denominator}} = \frac{476.0}{238.0} = 2.0
\][/tex]
5. Calculate the y-intercept [tex]\( c \)[/tex]:
The y-intercept can be found using the slope and the mean values calculated earlier:
[tex]\[
c = \overline{y} - m \cdot \overline{x} = 77.0 - 2.0 \cdot 15.0 = 77.0 - 30.0 = 47.0
\][/tex]
6. Form the linear equation [tex]\( y = mx + c \)[/tex]:
Using the calculated slope and intercept, the linear equation that represents the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is:
[tex]\[
y = 2.0x + 47.0
\][/tex]
Thus, the linear equation that gives the rule for the values in this table is:
[tex]\[
y = 2.0x + 47.0
\][/tex]
