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What is the ***constant of proportionality*** between dollars earned by the singer and dollars earned by selling the song?

Write an equation that relates dollars ($) earned by the singer, **y**, to dollars ($) earned from the sale of the song, x.

How much money did the singer earn when the store makes $400 in sales?

What does the point (0, 0) mean on this graph?

Let's analyze the graph and solve the problem step by step.

### 1. **Finding the constant of proportionality:**
   The constant of proportionality refers to how much the singer earns for each dollar made from selling the song. This is essentially the slope of the line, which can be calculated using any two points on the line.

From the graph, we have the points:
- $(200, 40)$
- $(600, 120)$
- $(1000, 200)$

We can calculate the slope (constant of proportionality) between any two points. Let's use $(200, 40)$ and $(1000, 200)$.

The formula for slope is:

\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the points $(200, 40)$ and $(1000, 200)$:

\[
\text{slope} = \frac{200 - 40}{1000 - 200} = \frac{160}{800} = 0.2
\]

So, the constant of proportionality is **0.2**.

### 2. **Equation relating dollars earned by the singer, $y$, to dollars earned from the song sales, $x$:**
   Since the constant of proportionality is 0.2, the equation is a linear one of the form:

\[
y = 0.2x
\]

where $y$ is the dollars earned by the singer and $x$ is the dollars earned from the song sales.

### 3. **How much did the singer earn when the store makes $400 in sales?**
   To find out how much the singer earned when the store made $400 in sales, substitute $x = 400$ into the equation $y = 0.2x$:

\[
y = 0.2 \times 400 = 80
\]

Thus, the singer earned **$80** when the store made $400 in sales.

### 4. **Meaning of the point $(0, 0)$ on the graph:**
   The point $(0, 0)$ means that if the store has made **$0 in sales**, the singer has earned **$0**. In other words, without any sales, the singer doesn't earn anything.

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Would you like further details on any part of this explanation?

Here are 5 related questions to deepen your understanding:

1. How would the equation change if the constant of proportionality were 0.3?
2. If the store made $500 in sales, how much would the singer earn based on the current equation?
3. What would the graph look like if the singer earned a fixed fee in addition to the proportional earnings?
4. Can you calculate the singer’s earnings if the store’s sales reached $1200?
5. What happens to the slope of the line if the singer earns a larger percentage of sales?

**Tip:** When interpreting graphs, the slope (constant of proportionality) helps determine how two quantities are related linearly.

What is the constant of proportionality between dollars earned by the singer and dollars earned by selling the song? Write an equation that relates dollars ($) earned by the singer, y, to dollars ($) earned from the sale of the song, x. How much money did the singer earn when the store makes $400 in sales? What does the point (0, 0) mean on this graph?

This lesson explores how to find the constant of proportionality between dollars earned by a singer and sales revenue. We provide a step-by-step approach to calculating the slope, forming the equation y = 0.2x, and interpreting key points on the graph, including (0, 0). Suitable for students in Grades 7-9.

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