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Algebraic Connections Exploration

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Algebraic Connections Exploration

A group is planning a car wash fundraiser. Use algebra to model costs and income, solve equations and inequalities, and explain how different representations connect. Show working where needed.

Build and connect models

Represent the fundraiser situation using algebraic expressions, equations and graphs.

1.Ruben estimates that supplies will cost $36, and the group will earn $8 for each car washed. Which expression gives the fundraiser’s profit, P dollars, after washing c cars?

P = 36 + 8c
P = 8c − 36
P = 36c − 8
P = 8(c − 36)

2.Oliver says the fundraiser breaks even when its profit is zero. Use the model P = 8c − 36 to find the number of cars needed to break even. Show an equation and solve it.

3.Ari graphs the equation P = 8c − 36, with c on the horizontal axis and P on the vertical axis. Which feature of the graph shows the break-even point, and what does it mean in this context?

4.The group wants to make at least $100 profit. Write and solve an inequality using P = 8c − 36. Then state the minimum whole number of cars it must wash.

5.Show the solution to 8c − 36 ≥ 100 on the number line. Use a closed dot because 17 is included, and shade towards larger values.

Investigate relationships

Compare models and check whether algebraic claims make sense.

6.Two local suppliers offer water blasters for the event. Supplier A charges $18 hire plus $3 per hour. Supplier B charges $6 hire plus $5 per hour. Let h be the number of hire hours. Write both cost rules and calculate when the costs are equal.

7.The cost rules in Question 6 are straight lines. State each line’s gradient and vertical intercept, then explain what their intersection represents in this situation.

8.A volunteer writes 3(2x − 4) = 6x − 4. Decide whether the statement is true or false. Correct it if needed, and name the algebraic property used.

TrueFalse

9.The fundraiser sells a wash-and-vacuum package for $15. Basic washes cost $8. The group earns $111 altogether from selling 11 packages and basic washes. If b is the number of basic washes, form an equation and find b.

10.A volunteer notices that Question 9 gives a non-whole-number answer. If all 11 sales are whole washes or packages, what total would make the numbers of each type whole? Give one possible total and explain.