Let's address the problems one by one:

7. Tennis Club Cost Problem

(a) Write an equation that describes Felicia's total cost as a function of the number of times she plays tennis.

• Initial membership fee: $125
• Cost per play: $5 per game
• Let $C$ be the total cost and $n$ be the number of times she plays tennis.

$C(n) = 125 + 5n$

(b) Describe the domain and range of the function.

• Domain: $n \geq 0$, since she cannot play a negative number of times.
• Range: $C \geq 125$, since the total cost starts at $125 and increases with the number of games.

(c) Felicia does not want to spend more than $275 to play tennis. What is the maximum number of times she can play?

$C(n) \leq 275$

Substitute the equation $125 + 5n \leq 275$:

$5n \leq 150$ $n \leq 30$

Thus, the maximum number of times she can play is 30.

8. Soccer Team Fundraising Problem

(a) Write a verbal model for the total amount $A$ raised by the soccer team.

The total amount $A$ is the sum of:

1. $n$, the money raised per kilometer walked ($92n$).
2. $d$, the additional donations not dependent on walking.

$A = 92n + d$

(b) The team walked 8 kilometers and raised $842. Is there enough information to find $d$?

Substitute $n = 8$ into $A = 92n + d$:

$842 = 92(8) + d$ $842 = 736 + d$ $d = 842 - 736 = 106$

Yes, the additional donations amount to $106.

9. Line Equation

Write the equation of a line passing through $(-7, -6)$ with slope $m = 4$.

The point-slope form of a line is:

$y - y_1 = m(x - x_1)$

Substitute $(-7, -6)$ and $m = 4$:

$y - (-6) = 4(x - (-7))$ $y + 6 = 4(x + 7)$ $y = 4x + 28 - 6$ $y = 4x + 22$

The equation is:

$y = 4x + 22$

10. Linear Relationship Between Price and Sales

(a) Create the equation for the relationship between price and sales.

The slope of the linear relationship is:

$m = \frac{\Delta y}{\Delta x} = \frac{900 - 1050}{9 - 5} = \frac{-150}{4} = -37.5$

Using the point-slope form, pick $(5, 1050)$:

$y - 1050 = -37.5(x - 5)$ $y - 1050 = -37.5x + 187.5$ $y = -37.5x + 1237.5$

(b) Predict the sales when the price is $12.

Substitute $x = 12$:

$y = -37.5(12) + 1237.5$ $y = -450 + 1237.5 = 787.5$

The predicted sales are approximately 788 gadgets.

Do you want further explanations for any of these problems?
Here are 5 related questions to deepen your understanding:

1. How would the cost equation change if the tennis club increased the fee per game?
2. What would the fundraising equation look like if there were more sponsors?
3. Can you graph the line $y = 4x + 22$ and identify its intercepts?
4. How does the slope affect the sales predictions for the gadget factory?
5. What happens to Felicia's maximum games if her budget increases by $50?

Tip: When solving linear equations, always double-check your substitution to avoid small calculation errors.