To create a linear equation that models the population increase of Arizona from 1980 to 2000, we need to determine the rate of change in population per year and the initial population value.
1. Given Data Points:
- In 1980, the population was 2.81 million.
- In 2000, the population was 4.88 million.
2. Calculate the Rate of Increase (Slope):
The slope (m) can be calculated using the formula for the rate of change between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[
m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Here:
[tex]\[
x_1 = 1980, \quad y_1 = 2.81, \quad x_2 = 2000, \quad y_2 = 4.88
\][/tex]
Substituting the values:
[tex]\[
m = \frac{4.88 - 2.81}{2000 - 1980} = \frac{2.07}{20} = 0.1035
\][/tex]
The slope (m) is [tex]\(0.1035\)[/tex] million people per year.
3. Determine the Y-Intercept:
The y-intercept (b) can be found using the linear equation [tex]\(y = mx + b\)[/tex]. We can use one of the given points, for example, (1980, 2.81).
Substitute [tex]\(x = 1980\)[/tex], [tex]\(y = 2.81\)[/tex], and [tex]\(m = 0.1035\)[/tex] into the equation:
[tex]\[
2.81 = 0.1035 \cdot 1980 + b
\][/tex]
Solve for [tex]\(b\)[/tex]:
[tex]\[
2.81 = 204.93 + b
\][/tex]
[tex]\[
b = 2.81 - 204.93 = -202.12
\][/tex]
The y-intercept (b) is approximately [tex]\(-202.12\)[/tex] million.
4. Form the Linear Equation:
Now, combine the slope and y-intercept to construct the linear equation in the form [tex]\(y = mx + b\)[/tex]:
[tex]\[
y = 0.1035x - 202.12
\][/tex]
Therefore, the linear equation that models the population increase in Arizona from 1980 to 2000 is:
[tex]\[
y = 0.1035x - 202.12
\][/tex]
