Sure, let's derive the equation of the line that best represents the given data points.
Given data points are:
[tex]\[
\begin{array}{c|r}
x & y \\
\hline
-3 & -5 \\
-2 & -3 \\
-1 & -1 \\
0 & 1 \\
1 & 3 \\
2 & 5 \\
\end{array}
\][/tex]
To find the equation of a line in the form [tex]\( y = mx + b \)[/tex], we need to determine the slope ([tex]\(m\)[/tex]) and the y-intercept ([tex]\(b\)[/tex]).
1. Calculate the slope ([tex]\(m\)[/tex]):
The slope ([tex]\(m\)[/tex]) of the line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Using the points [tex]\((-3, -5)\)[/tex] and [tex]\( (2, 5) \)[/tex]:
[tex]\[
x_1 = -3, \quad y_1 = -5, \quad x_2 = 2, \quad y_2 = 5
\][/tex]
Substitute these values into the slope formula:
[tex]\[
m = \frac{5 - (-5)}{2 - (-3)} = \frac{5 + 5}{2 + 3} = \frac{10}{5} = 2.0
\][/tex]
So, the slope [tex]\(m\)[/tex] is 2.0.
2. Calculate the y-intercept ([tex]\(b\)[/tex]):
Using the slope-intercept form [tex]\( y = mx + b \)[/tex], we can find [tex]\(b\)[/tex] by substituting one of the points and the slope.
Using the point [tex]\((-3, -5)\)[/tex] and the slope [tex]\(m = 2.0\)[/tex]:
[tex]\[
-5 = 2.0 \times (-3) + b
\][/tex]
Solve for [tex]\(b\)[/tex]:
[tex]\[
-5 = -6.0 + b \implies b = -5 + 6.0 \implies b = 1.0
\][/tex]
Therefore, the y-intercept [tex]\(b\)[/tex] is 1.0.
Complete equation:
Combining the slope [tex]\(m = 2.0\)[/tex] and the y-intercept [tex]\(b = 1.0\)[/tex], the equation of the line is:
[tex]\[
y = 2.0x + 1.0
\][/tex]
