053i - Solving Linear Models

Lesson 53 — Solving Applied Linear Problems in Financial Contexts

Strand: Algebra | Descriptor: AC9M8A03 | Duration: 45 minutes

Learning Intentions

• To solve linear equations that arise from financial models, both by evaluating for a known input and by rearranging to find an unknown input.
• To find the break-even point between two competing linear models by setting them equal and solving.

Success Criteria

I can:

1. Evaluate a linear model to find the output for a given value of the independent variable.
2. Solve a linear equation to find the input value that produces a target output.
3. Set two linear expressions equal to each other and solve to find where two models agree.
4. Decide whether a question calls for evaluating or solving, and justify the choice.

Warmup

(5 minutes — quick-fire, mini whiteboards)

Recall the modelling cycle from Lesson 51:

Stage	What happens
1. Formulate	Translate the real situation into mathematics.
2. Represent	Choose a table, graph, equation or description.
3. Solve	Use algebra or graphing to answer the mathematical question.
4. Interpret	Translate the mathematical answer back into the real situation.
5. Evaluate	Check whether the model is reasonable, and refine it if needed.

Today we focus on Stage 3: Solve. (Lessons 54 and 55 take Stages 4 and 5; Lesson 56 puts the whole cycle together.)

1. Bolt Mobile: . Find when .
2. FitZone: . Estimate how many whole weeks 200$ buys, before calculating exactly.

Discussion: Question 1 gave you the input and asked for the output — that’s evaluating. Question 2 gave you the output ($200) and asked for the input — that’s solving. Both start from the same equation; only the direction of travel differs.

Activities

Activity 1 — Explicit Instruction: Evaluating versus Solving (12 min)

I do: Courier model from Lesson 51 enrichment: , where is mass in kg.

Evaluate for :

A kg parcel costs 17$.

Solve for given :

A 3515$ kg parcel.

We do: FitZone gym, . Together: evaluate for ; then solve for given .

You do:

1. Plumber: . Evaluate for . Then solve for given .
2. Taxi: . Evaluate for . Then solve for given .

Activity 2 — Explicit Instruction: Solving for the Break-even Point (12 min)

Recall from Lesson 52 that two competing models plotted on the same axes appear to cross at a break-even point — the input value where both give the same output. Today we find that point exactly, by solving.

I do: Rideshare companies from Lesson 52: Company A , Company B .

The lines cross at km — consistent with the Lesson 52 graph, which showed the crossing between and . Solving gives the exact value that the graph could only estimate.

We do: Together, phone plans and .

You do: Car hire from Lesson 52: Option A (no flat fee), Option B . Solve for the break-even value of .

Activity 3 — Inquiry Task: Comparing Two Lawn-mowing Businesses (11 min)

Pairs, then whole-class share.

GreenCut charges a 35$0.50$20$0.65250\ \text{m}^2$ lawn.

Find, algebraically, the lawn area at which the two businesses charge the same amount. Then determine which business is cheaper for the homeowner’s lawn.

Socratic scaffolding:

Prompt	Purpose
Understand: what does “charge the same amount” mean mathematically?	The two cost expressions are equal — set them equal to each other.
What is the independent variable here?	Lawn area, , in square metres.
Devise a plan	Write both cost expressions, set them equal, and solve for .
Carry out the plan	.
Look back — does this answer the homeowner’s actual question?	Not yet — it gives the break-even area, not which business suits her specific lawn. A further step is needed.
How do you use the break-even value to answer the real question?	Compare to the break-even area, and check which rate applies beyond it.

At the two businesses cost the same. Beyond , QuickMow’s steeper rate (0.65$0.50250\ \text{m}^2 > 100\ \text{m}^2$, GreenCut is cheaper. Check:

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Given , evaluate for .
2. Given , solve for given .
3. Two market stalls charge and . Solve for the break-even value of .
4. Reasoning. A student is asked, “How much would a kg parcel cost?” and also “What mass of parcel costs 41$?” Explain, without calculating, which question requires evaluating and which requires solving.

Answers: 1. ; 2. ; 3. ; 4. The first gives the input (mass) and asks for the output (cost) — evaluate; the second gives the output (cost) and asks for the input (mass) — solve.

Common Misconceptions

Misconception	How to pre-empt it
”Solving” and “evaluating” are the same thing.	Explicitly name the direction of travel each time: known input → output (evaluate); known output → input (solve).
When solving, students substitute the target value into the wrong side of the equation.	Model writing the full equation first (e.g. ) before doing anything else.
At a break-even point, students solve but forget it is a value of the independent variable, not the cost.	After solving, ask “what does this number actually count?” before moving on.
Sign errors when collecting variable terms on one side (e.g. vs ).	Rewrite the step explicitly: “subtract the smaller coefficient term from both sides,” and check the sign every time.
Recurring decimals (e.g. ) are rounded incorrectly or dropped.	Discuss appropriate rounding in context immediately — this is the bridge to Lesson 54’s focus on interpretation.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). Two phone plans are and , where is data in MB. For what value of do the two plans cost the same?

Answer

E2 (Kangaroo style). A tank contains L and drains at L per minute: . Solve for the time at which the tank is exactly one-third full.

Answer

One-third of L is L.

E3 (Challenge). A courier charges . A rival courier charges . Solve for the mass at which the rival becomes more expensive than the original.

Answer

Break-even: kg. For masses greater than kg, the rival’s steeper rate (3.50$2$/kg) makes it more expensive.

Homework

1. Given , evaluate for .
2. Given , solve for given .
3. A caterer charges . Solve for the number of guests, , if the total bill was 472$.
4. Two savings plans are and . Solve for the value of at which the two balances are equal.
5. Reasoning. Explain, in your own words, why solving an equation such as can be thought of as “undoing” the steps used to evaluate the expression, in reverse order.
6. Challenge. A parking station charges . A second station charges a flat 293$-hour stay.

Answers: 1. ; 2. ; 3. ; 4. ; 5. Evaluating starts with and applies "" then ""; solving starts with the result and reverses those steps in the opposite order: "" then ""; 6. ; for a -hour stay, station 1 costs 17 < $29$, so the hourly station is cheaper.