Sure, let's work through the problem step by step using the data provided about Alyssa's daily earnings and total sales.
Given the table:
[tex]\[
\begin{array}{|r|r|}
\hline \text{Total Sales (x)} & \text{Daily Earnings (y)} \\
\hline 0 & 80 \\
\hline 200 & 110 \\
\hline 500 & 155 \\
\hline 800 & 200 \\
\hline 1200 & 260 \\
\hline
\end{array}
\][/tex]
To determine the equation of the line that represents Alyssa's daily earnings [tex]\( y \)[/tex] based on her total sales [tex]\( x \)[/tex], follow these steps:
1. Identify two points to calculate the slope:
- The first point is (0, 80).
- The second point is (1200, 260).
2. Calculate the slope (m) using the formula for slope between two points [tex]\((x1, y1)\)[/tex] and [tex]\((x2, y2)\)[/tex]:
[tex]\[
m = \frac{y2 - y1}{x2 - x1}
\][/tex]
Substituting the provided points:
[tex]\[
m = \frac{260 - 80}{1200 - 0} = \frac{180}{1200} = 0.15
\][/tex]
3. Determine the y-intercept (b), which is the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex].
- In this case, the first point directly gives us the y-intercept because [tex]\( x = 0 \)[/tex] and [tex]\( y = 80 \)[/tex].
4. Formulate the equation of the line which goes through these points. The general form of the linear equation is:
[tex]\[
y = mx + b
\][/tex]
Substituting the slope (m) and the y-intercept (b):
[tex]\[
y = 0.15x + 80
\][/tex]
Summarizing these steps and filling in the blanks for the question:
1. The equation that gives Alyssa's daily earnings based on her total sales is:
[tex]\[
y = 0.15x + 80
\][/tex]
2. When this data is plotted on a graph with a line connecting all the points, the slope of the equation is:
[tex]\[
0.15
\][/tex]
3. The y-intercept of the line represents Alyssa's initial daily earnings (when total sales are zero), which in this scenario is:
[tex]\[
80
\][/tex]
