To determine the linear equation that describes the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] based on the given table, we follow these steps:
1. Identify two points from the table:
Given the points are:
[tex]\[
(2, 58), (3, 65), (4, 72), (5, 79)
\][/tex]
2. Calculate the slope (m):
The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Using the points [tex]\((2, 58)\)[/tex] and [tex]\((5, 79)\)[/tex]:
[tex]\[
m = \frac{79 - 58}{5 - 2} = \frac{21}{3} = 7.0
\][/tex]
3. Calculate the y-intercept (b):
The formula for the y-intercept [tex]\( b \)[/tex] when the slope [tex]\( m \)[/tex] and a point [tex]\((x_1, y_1)\)[/tex] are known is:
[tex]\[
b = y_1 - m \cdot x_1
\][/tex]
Using the point [tex]\((2, 58)\)[/tex] and the slope [tex]\( m = 7.0 \)[/tex]:
[tex]\[
b = 58 - (7.0 \cdot 2) = 58 - 14 = 44.0
\][/tex]
4. Form the linear equation:
Using the slope [tex]\( m = 7.0 \)[/tex] and the y-intercept [tex]\( b = 44.0 \)[/tex], the linear equation in the form [tex]\( y = mx + b \)[/tex] is:
[tex]\[
y = 7.0x + 44.0
\][/tex]
So, the linear equation that describes the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the given table is:
[tex]\[
y = 7.0x + 44.0
\][/tex]
