To find the linear equation that fits the given data points, we can follow these steps:
1. Identify Data Points:
We have four data points:
[tex]\[
(-66, -82), (-27, -43), (12, -4), (51, 35)
\][/tex]
2. Calculate the Slopes:
We calculate the slopes ([tex]\(m\)[/tex]) between successive points using the formula:
[tex]\[
m_{ij} = \frac{y_j - y_i}{x_j - x_i}
\][/tex]
Applying the formula between the first point [tex]\((-66, -82)\)[/tex] and the rest:
[tex]\[
m_{12} = \frac{-43 - (-82)}{-27 - (-66)} = \frac{-43 + 82}{39} = \frac{39}{39} = 1
\][/tex]
[tex]\[
m_{13} = \frac{-4 - (-82)}{12 - (-66)} = \frac{-4 + 82}{78} = \frac{78}{78} = 1
\][/tex]
[tex]\[
m_{14} = \frac{35 - (-82)}{51 - (-66)} = \frac{35 + 82}{117} = \frac{117}{117} = 1
\][/tex]
3. Average Slope:
The slopes calculated are all the same:
[tex]\[
m = 1
\][/tex]
4. Calculate the Y-Intercept:
Using the slope [tex]\(m = 1\)[/tex] and one of the given points, for example [tex]\((-66, -82)\)[/tex], we can find the y-intercept [tex]\(b\)[/tex] using the equation [tex]\(y = mx + b\)[/tex].
[tex]\[
-82 = 1(-66) + b
\][/tex]
[tex]\[
-82 = -66 + b
\][/tex]
[tex]\[
b = -82 + 66
\][/tex]
[tex]\[
b = -16
\][/tex]
5. Form the Linear Equation:
Substituting [tex]\(m = 1\)[/tex] and [tex]\(b = -16\)[/tex] into the linear equation [tex]\(y = mx + b\)[/tex]:
[tex]\[
y = 1x + (-16) \quad \text{or} \quad y = x - 16
\][/tex]
Thus, the linear equation that describes the data is:
[tex]\[
y = x - 16
\][/tex]
