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Algebraic Relationships · L2 of 6 ~50 min MST-12-S2-01 ⚡ +90 XP available

Modelling with Simultaneous Equations & Break-Even Analysis

Simultaneous equations are the engine behind every break-even analysis in business. You set up cost and revenue as two linear equations, then find where they intersect — the break-even point where profit switches from negative to positive.

Today's hook — A food stall costs $400/day to operate and sells items at $8 each. When does the stall start making a profit? The answer is one simultaneous equation away.
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Recall — your gut answer first
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A food stall has fixed costs of $400 per day plus $2 per item in ingredients. Items sell for $8 each. How many items must the stall sell to cover costs exactly?

Without calculating — write a guess and how you'd think about it. What two quantities would you need to make equal?

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The key idea you need to own
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A worded problem with two unknowns can always be modelled as two simultaneous equations. The key step is converting the words into algebra — then the methods from Lesson 1 take over.

For a break-even problem: write a cost equation ($C = \ldots$) and a revenue equation ($R = \ldots$), both in terms of units sold. At break-even, $C = R$. Solving gives the break-even quantity, and substituting gives the break-even dollar value.

Break-even: $\text{Revenue} = \text{Cost}$  →  $\text{Profit} = 0$
Define variables first
Before writing any equation, clearly define what $x$ and $y$ (or named variables) represent, including units. This prevents errors and earns method marks.
Check your equations make sense
Does the cost increase as units increase? Does revenue start at zero (or a fixed amount)? Substitute in easy values to sanity-check before solving.
Spreadsheets = quick check
NESA expects you to use a spreadsheet to find or verify the break-even point. Set up a table of $n$ values with cost and revenue columns — break-even is where they cross.
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What you'll master
Know

Key facts

  • A practical problem with two unknowns yields two simultaneous equations
  • Break-even occurs where cost equals revenue: $C = R$
  • Profit $= R - C$; profit is positive above break-even, negative below
  • Spreadsheets can be used to identify or verify the break-even point
Understand

Concepts

  • How to read a worded problem and extract two linear equations
  • Why the break-even point is the intersection of cost and revenue lines
  • The difference between fixed costs, variable costs, and revenue
  • How profit and loss relate to the position relative to break-even
Can do

Skills

  • Define variables and write two simultaneous equations from a worded problem
  • Solve the system to find the answer in context
  • Identify break-even quantity and dollar value for a linear cost/revenue model
  • Use a spreadsheet to find or verify break-even
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Key terms
Fixed costA cost that does not change with the number of units produced or sold (e.g. rent, equipment, daily wage).
Variable costA cost that changes in proportion to units produced (e.g. materials per item, ingredients per meal).
RevenueTotal income from sales: $R = \text{selling price} \times n$, where $n$ is the number of units sold.
Total cost$C = \text{fixed cost} + \text{variable cost per unit} \times n$. This is a linear function of $n$.
Break-even pointThe number of units (and corresponding dollar value) at which total revenue equals total cost — profit is zero.
Profit / LossProfit $= R - C$. Positive = profit; negative = loss. At break-even, profit $= 0$.
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Setting Up Simultaneous Equations from a Worded Problem
core concept

Every worded problem that can be solved with simultaneous equations contains two pieces of information — two relationships — each of which becomes one equation. The art is identifying which quantities are related and how.

A reliable strategy:

  1. Read carefully and underline the two unknown quantities. These become your variables (e.g. $x$ = number of adult tickets, $y$ = number of child tickets).
  2. Find the two conditions. Look for two sentences that describe a total or a constraint. Each becomes one equation.
  3. Translate into algebra. Write an equation for each condition using your variables.
  4. Solve and interpret. Solve the system, then write the answer in the context of the problem — include units and check it makes practical sense.
Typical worded problem structure: Problems often give (i) a total count: $x + y = k$, and (ii) a total value: $ax + by = m$. These two equations, when solved, give the exact split between the two quantities.
Answer in context: The final answer must be written in the context of the problem. "The solution is $x = 120$" is incomplete. "The company must sell 120 units to break even" earns full marks.
What to write in your book
  • Define variables clearly: "Let $x$ = …" before writing any equation.
  • Two conditions → two equations. Look for a total count and a total value.
  • Always answer in context: include the unit and the meaning of the variable.

Quick check: A problem states "there are 30 animals, some chickens and some cows, with 86 legs in total." Which pair of equations correctly models this?

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Break-Even Analysis — Cost and Revenue Equations
core concept

In any business, the break-even point is where you stop losing money and start making it. Mathematically, it is the value of $n$ (units sold) where the revenue line crosses the cost line — the intersection of two linear equations.

Setting up a break-even model:

  • Total cost: $C = F + vn$, where $F$ is the fixed cost (e.g. setup, rent) and $v$ is the variable cost per unit (e.g. cost of materials per item).
  • Revenue: $R = pn$, where $p$ is the selling price per unit.
  • Break-even: Set $C = R$ and solve for $n$. The result is the break-even quantity.
  • Dollar value at break-even: Substitute the break-even $n$ into either equation to find the corresponding cost/revenue dollar amount.
Food stall example: Fixed cost $F = \$400$, variable cost $v = \$2$/item, selling price $p = \$8$/item.
Cost: $C = 400 + 2n$   Revenue: $R = 8n$
Break-even: $8n = 400 + 2n \Rightarrow 6n = 400 \Rightarrow n = 66.\overline{6}$. Since items are whole, the stall must sell 67 items to first exceed costs. Dollar value: $R = 8 \times 67 = \$536$.
Interpreting above and below break-even: When $n < $ break-even quantity, $C > R$ — the business makes a loss. When $n >$ break-even, $R > C$ — the business makes a profit. The profit at any quantity is $P = R - C = (p - v)n - F$.
What to write in your book
  • $C = F + vn$ (cost = fixed + variable × units). $R = pn$ (revenue = price × units).
  • Break-even: set $C = R$ and solve for $n$. Round up to next whole item if $n$ is not an integer.
  • Profit $= R - C$ — positive above break-even, negative below.

True or false: If the break-even quantity works out to 83.4 items, the business should sell 83 items to first make a profit (round down).

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Using a Spreadsheet to Find the Break-Even Point
core concept

NESA explicitly requires you to find and verify the break-even point using a spreadsheet. This is not just a checking tool — it is a required method you may be assessed on.

Setting up a spreadsheet for break-even:

  1. Column A: Units produced/sold ($n$ = 0, 10, 20, … or 1, 2, 3, … depending on the scale)
  2. Column B: Total cost — formula =F + v*A2 (replacing $F$ and $v$ with numbers)
  3. Column C: Revenue — formula =p*A2
  4. Column D (optional): Profit — formula =C2-B2
  5. Find break-even: Look for the row where cost and revenue are equal, or where profit changes from negative to positive (zero is the break-even).
Profit and loss with a spreadsheet: If column D shows profit, the break-even is where D changes sign (from negative to positive). You can then use the exact values to calculate the break-even quantity and the dollar value. For non-integer break-even, the first row where profit ≥ 0 is the first profitable level.
HSC exam tip: When a question says "use a spreadsheet to identify the break-even point", show your column headings, at least 3 rows of values around the break-even, and state the break-even quantity and cost/revenue amount.
What to write in your book
  • Spreadsheet: Column A = units; Col B = cost formula; Col C = revenue formula; Col D = profit = revenue − cost.
  • Break-even = row where profit changes from negative to zero or positive.
  • Always state the break-even quantity and the corresponding dollar value.

Match each spreadsheet column to its formula/meaning:

Worked examples · 3 problems

PROBLEM 1 · SETTING UP A WORDED PROBLEM

A theatre sells adult tickets for $22 and child tickets for $12. One night, 350 tickets were sold for total revenue of $5,900. How many adult and child tickets were sold?

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Define variables
Let $a$ = number of adult tickets, $c$ = number of child tickets
Two unknowns → two variables. Always state what each variable represents.
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PROBLEM 2 · BREAK-EVEN ANALYSIS

A market stall has fixed costs of $180/day and spends $4 per item on materials. Items are sold for $10 each. Find (a) the break-even quantity and (b) the profit if 50 items are sold.

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Set up equations (let $n$ = items sold)
Cost: $C = 180 + 4n$
Revenue: $R = 10n$
Fixed cost is $180 (does not depend on $n$). Variable cost is $4 per item. Revenue is $10 per item sold.
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PROBLEM 3 · SPREADSHEET INTERPRETATION

A small business sets up a spreadsheet to model cost and revenue. The table shows: at $n = 80$, cost = $1,040, revenue = $960; at $n = 90$, cost = $1,080, revenue = $1,080. Identify the break-even point and verify algebraically if the cost equation is $C = 800 + 3n$ and revenue is $R = 12n$.

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From the table, identify break-even
At $n = 80$: $C = 1040 > R = 960$ (loss)
At $n = 90$: $C = R = 1080$ (break-even)
The break-even point is where cost equals revenue in the table. It occurs at $n = 90$, with a cost/revenue of $1,080.
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Top 3 list: List THREE things you must state in your answer when a question asks you to "find the break-even point."

Quick-fire practice · 3 problems

1

A festival stall has fixed costs of $250/day. Each item costs $3 to produce and is sold for $8. Find the break-even quantity.

2

A parking station has $500 in fixed costs per day. Each car pays $6 to park. If the variable operating cost per car is $0.50, how many cars are needed to break even?

3

A store sells two types of juice: mango ($3.50) and orange ($2.00). On Monday 80 bottles were sold for $214. How many of each type were sold?

Fill the blanks: In a break-even model, the cost equation is $C = F + vn$ where $F$ is the cost and $v$ is the cost per unit. At break-even, profit equals .

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Revisit your thinking

Food stall: $C = 400 + 2n$, $R = 8n$. Break-even: $8n = 400 + 2n \Rightarrow 6n = 400 \Rightarrow n = 66.\overline{6}$. The stall must sell at least 67 items to turn a profit. Was your initial estimate close?

The key insight: break-even is driven by the gap between selling price and variable cost ($8 - \$2 = \$6$ per item). The fixed cost is recovered at a rate of $6 per item — $400 \div \$6 = 66.\overline{6}$ items.

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Short answer — exam-style questions
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ApplyBand 33 marks

SA 1. A school canteen sells pies for $4.50 and sandwiches for $3.00. On Friday, 120 items were sold for a total of $459. How many of each item were sold? (3 marks)

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ApplyBand 44 marks

SA 2. A craft business has weekly fixed costs of $600. Each item costs $12 in materials and is sold for $30. (a) Write equations for weekly cost $C$ and revenue $R$ in terms of items sold $n$. (b) Find the break-even quantity. (c) Find the profit if 45 items are sold in a week. (4 marks)

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AnalyseBand 54 marks

SA 3. A spreadsheet for a food van shows: at $n = 40$ items, cost = $560 and revenue = $440 (loss of $120); at $n = 60$, cost = $680 and revenue = $660 (loss of $20); at $n = 70$, cost = $740 and revenue = $770 (profit of $30). (a) Between which two values of $n$ does break-even occur? (b) If the cost equation is $C = 400 + 5n$ and revenue is $R = 11n$, find the exact break-even quantity and verify it is consistent with the spreadsheet. (4 marks)

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📖 Comprehensive answers (click to reveal)

Drill 1: $C = 250 + 3n$, $R = 8n$. $8n = 250 + 3n \Rightarrow 5n = 250 \Rightarrow n = 50$ items.

Drill 2: $C = 500 + 0.5n$, $R = 6n$. $6n = 500 + 0.5n \Rightarrow 5.5n = 500 \Rightarrow n = 90.\overline{9} \approx 91$ cars.

Drill 3: Let $m$ = mango, $o$ = orange. $m+o=80$ and $3.5m+2o=214$. Sub $m=80-o$: $3.5(80-o)+2o=214 \Rightarrow 280-1.5o=214 \Rightarrow o=44$, $m=36$.

SA 1 (3 marks): Let $p$ = pies, $s$ = sandwiches. $p+s=120$ [1] and $4.5p+3s=459$ [1]. Sub $p=120-s$: $4.5(120-s)+3s=459 \Rightarrow 540-1.5s=459 \Rightarrow s=54$, $p=66$ [1]. Verify: $4.5(66)+3(54)=297+162=459$ ✓.

SA 2 (4 marks): (a) $C=600+12n$, $R=30n$ [1]. (b) $30n=600+12n \Rightarrow 18n=600 \Rightarrow n=33.\overline{3}$, so must sell 34 items [1]. (c) At $n=45$: $R=1350$, $C=600+540=1140$, Profit $=\$210$ [1 for each correct value, or 1 mark for correct method + 1 for answer].

SA 3 (4 marks): (a) Between $n=60$ and $n=70$ (profit changes from negative to positive) [1]. (b) Set $C=R$: $400+5n=11n \Rightarrow 400=6n \Rightarrow n=66.\overline{6}$ [1]. Must sell 67 items. Consistency: at $n=67$, $C=400+335=735$, $R=737$, profit=$+\$2$ (positive for first time) — consistent with break-even between 60 and 70 ✓ [1]. State break-even quantity = 67 items [1].

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Boss battle · The Business Analyst
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Five timed break-even and modelling questions. Gold tier requires 90% accuracy and speed.

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Algebraic Relationships · Maths Standard