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## Sagot :

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]