Comparing Linear Models and Break-Even Points

Compare options using linear equations, find where two models are equal, and explain which option is better under different conditions.

45 min Algebra Linear relationships Lesson 13 of 13
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Choose how you work: type answers on screen, or work in your book.

Printable worksheet

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Use the printable version for comparing linear models, solving break-even points and writing justified decisions.

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Think First

Plan A costs $20 plus $5 per gigabyte. Plan B costs $50 plus $2 per gigabyte. Which plan is cheaper?

Type what extra information you need before deciding.

Write what extra information you need before deciding.

Write your response in your book
Saved

Know

  • A break-even point is where two models have the same output.
  • The intersection point can represent equal cost, equal distance or equal savings.
  • The cheaper option can change depending on the input value.

Understand

  • A lower starting cost is not always the best long-term option.
  • A lower rate becomes more important as the input increases.
  • A decision should be justified for a specific range or condition.

Can Do

  • Write two linear equations for competing options.
  • Find and interpret a break-even point.
  • Justify which model is better before and after the break-even point.
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Break-Even Method

Model A = Model B
Set the two equations equal to find the input where outputs match.
$20 + 5g = 50 + 2g$
Solve for $g$, then interpret the result.

Core Content

1. Compare Both Starting Cost and Rate

A linear model with a low starting value can become expensive if its rate is high.

When comparing options, identify the fixed cost and the repeated rate for each option. Then compare at the input value that matters.

Common error: Do not choose the cheaper starting cost without considering the rate.
Worked Example 1

Find a phone-plan break-even point

Plan A costs $20 plus $5 per gigabyte. Plan B costs $50 plus $2 per gigabyte. Find the number of gigabytes where the plans cost the same.

Let $g$ be the number of gigabytes.

Plan A: $A = 20 + 5g$

Plan B: $B = 50 + 2g$

Set them equal: $20 + 5g = 50 + 2g$

$3g = 30$, so $g = 10$.

Interpretation: At 10 GB, both plans cost the same.

Worked Example 2

Decide which option is cheaper

Using the same plans, decide which is cheaper before and after 10 GB.

At 5 GB: Plan A costs $20 + 5(5) = 45$. Plan B costs $50 + 2(5) = 60$. Plan A is cheaper.

At 15 GB: Plan A costs $20 + 5(15) = 95$. Plan B costs $50 + 2(15) = 80$. Plan B is cheaper.

Conclusion: Plan A is cheaper below 10 GB. Plan B is cheaper above 10 GB.

Worked Example 3

Compare savings plans

Sam starts with $200 and saves $30 per week. Alex starts with $80 and saves $50 per week. When will they have the same amount?

Sam: $S = 200 + 30w$

Alex: $A = 80 + 50w$

$200 + 30w = 80 + 50w$

$120 = 20w$, so $w = 6$.

Interpretation: After 6 weeks, both people have the same savings.

Decision habit: A break-even point is only useful if you explain what happens before and after it.

2. A Table Can Check the Intersection

Gigabytes51015
Plan A: $20 + 5g$45$70$95
Plan B: $50 + 2g$60$70$80

The table confirms the break-even point at 10 GB because both costs are $70.

Activity

Compare Linear Models

  1. Company A charges $40 plus $12 per hour. Company B charges $70 plus $6 per hour. Find the break-even time.
  2. Decide which company is cheaper for 3 hours and for 8 hours.
  3. Two savings plans are $150 + 20w$ and $30 + 35w$. Find when they are equal.
  4. Explain why the cheaper option can change as the input increases.
Complete the model comparison practice in your book.

Revisit the Phone Plans

The plans break even at 10 GB. Below 10 GB, Plan A is cheaper. Above 10 GB, Plan B is cheaper because it has the lower per-gigabyte rate.

Explain the decision in your book.

Assessment

MC

Multiple Choice

Random questions from the lesson bank - feedback appears immediately.

SA

Short Answer

Find break-even points and justify decisions.

1. Company A charges $35 plus $10 per hour. Company B charges $65 plus $4 per hour. Find the break-even time. 4 MARKS

Answer in your book.

2. Using Question 1, decide which company is cheaper for 3 hours and for 8 hours. 4 MARKS

Answer in your book.

3. Explain what a break-even point means in a model comparison question. 2 MARKS

Answer in your book.

Break-Even Sort

Compare fixed costs and rates, solve the equality, then state which option wins on each side.

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