Picture selling notebooks at a table. Before you sell anything, you pay for the table. Each notebook also costs money to make. Cost is all the money you spend. Revenue is all the money customers pay you. The break-even point is where those two amounts match: sales have paid for the expenses, and no profit remains yet. Profit means revenue minus cost, the amount left after expenses. On a graph, quantity goes across and dollars go up. The cost and revenue lines meet at break-even. Their common height is the dollar amount, while the across coordinate tells you how many items you must sell.
The shared point is where the cost and revenue heights are equal.
Reminder
Linear model. A $10 fixed fee plus $3 per item is 10 + 3x.
Decimal division. 25,000 ÷ 0.01 = 2,500,000 because multiplying both numbers by 100 makes the divisor 1.
Systems. C(x) = R(x) selects a shared output at the same input.
Substitution. R(5) = 5 × 5 = 25 when the selling-price rule is R(x) = 5x.
Parallel lines. Equal rates and different starting fees keep the same gap at every input.
Signed subtraction. 20 − 22 = −2, a deficit of 2; 15 − 14 = 1, an excess of 1.
Why it works. If fixed cost is F and each item costs v to produce, total cost is C(x) = F + vx: the starting expense plus x repeated item expenses. If each sells for p, revenue is R(x) = px. Equating them gives F = (p − v)x. The difference p − v is each sale's contribution toward the fixed expense. Once enough contributions cover F, cost equals revenue. A positive fixed cost cannot be covered by sales when p ≤ v, because each contribution is zero or negative.
RuleRule: C(x) = F + vx, R(x) = px, and profit P(x) = R(x) − C(x). Break-even satisfies C(x) = R(x). For F > 0 and p > v, x = Fp − v.
The same idea, five ways
Say it
Say: find how many sales pay for all the expenses.
Write it
At the break-even point, revenue equals cost and profit is zero.
In math
C(x) = F + vx
R(x) = px
C(x) = R(x)
P(x) = R(x) − C(x)
x = Fp − v, p > v
Like
Each sale puts a little money toward paying back the table fee.
See itThe shared point is where the cost and revenue heights are equal.
The same idea, other ways
As a shared height
At break-even, the cost and revenue graphs have the same dollar height at the same quantity.
The shared point is where the cost and revenue heights are equal.
As repeated contributions
If an item sells for $5 and costs $3 to make, each sale leaves $2 toward a $10 setup fee. Five sales contribute 5 × $2 = $10.
Per-sale contributions cover the fixed cost.
.1Cost function
Fixed cost is a bill you pay once even if you make no items. Variable cost changes with quantity, such as buying paper for each notebook. A cost function adds those two expenses. In a linear model, per-item production cost is the cost slope, and fixed cost is the initial value.
Rule: C(x) = F + vx.
Rule: Fixed cost F is paid at quantity 0; variable cost vx grows with quantity.
Rule: Production models usually use x ≥ 0, with whole-number quantities for individual items.
The cost line starts at $10.
Reminder
Order of operations. 10 + 3 × 4 = 10 + 12 = 22.
Initial value. C(0) = 10 is the starting expense before production.
The same idea, five ways
Say it
Say: starting bill plus the bill for each item.
Write it
Total cost combines fixed and variable cost.
In math
fixed cost: F
variable cost: vx
C(x) = F + vx
C(0) = F
Like
The table fee plus the paper bill for your notebooks.
See itCost starts at the fixed expense, then rises by the per-item amount.
Worked exampleBuild a small cost model
You need the cost rule for a $10 setup fee and $3 per item, then the cost of 4 items.
The cost point combines the setup bill and four item expenses.
Write C(x) = 10 + 3x.The fixed cost is paid once, and per-item cost repeats x times.
C(4) = 10 + 3 × 4 = 22.Four item expenses total 12 before adding the setup fee.
Answer
C(x) = 10 + 3x.
C(4) = $22.
Check The separate bills are $10 and $12, and $10 + $12 = $22.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: C(x) = (10 + 3)x when setup is $10 and each item costs $3.
That formula charges the setup fee once per item instead of once total. At 4 items it gives $52.
✓ Instead: Use C(x) = 10 + 3x, which gives $22 at 4 items.
Tips and tricks
Tip: The fixed bill belongs outside the multiplication by quantity.
.2Revenue function
Revenue is the money coming in from sales before subtracting expenses. If each item sells for the same price, multiply the number sold by that price. Receiving money is different from keeping money; some of the revenue pays the cost.
Rule: R(x) = px when each of x items sells for price p.
Rule: In this model R(0) = 0 because no sales bring in no money.
Revenue has zero starting value in this sales model.
Reminder
Repeated addition. 5 × 4 is four payments of 5.
The same idea, five ways
Say it
Say: sale price times number sold.
Write it
Revenue records total sales money before expenses.
In math
R(x) = px
R(0) = 0
Like
The contents of the payment box before paying the bills.
See itRevenue starts at 0 and increases by $5 for each sale.
Worked exampleSeparate sales money from profit
You need revenue from selling 4 items at $5 each.
The revenue height gives sales money, not the remaining profit.
Write R(x) = 5x.The same selling price repeats for each item.
R(4) = 5 × 4 = 20.This is sales money before paying production or setup expenses.
Answer
R(x) = 5x.
R(4) = $20.
Check Adding four $5 payments gives $20, the same as multiplication.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: $20 of revenue means $20 of profit.
Profit must subtract the cost. If cost is $22, the remaining amount is negative.
✓ Instead: Profit is $20 − $22 = −$2, meaning a $2 loss.
Tips and tricks
Tip: Revenue means money in; profit means money left after cost.
.3Break-even, profit and loss
At break-even, all the sales money pays all the bills. Profit is zero. If revenue is larger, the difference is profit. If revenue is smaller, the negative difference records a loss. For item sales, use only quantities that make sense in the situation. Sometimes the graphs meet between two whole counts. You cannot sell part of an individual item, so, when each sale brings in more than it costs to make, the first whole count above that crossing is the first that covers the cost.
Rule: Solve C(x) = R(x) to find break-even.
Rule: P(x) > 0 means profit; P(x) < 0 means loss; P(x) = 0 means break-even.
Rule: With a positive fixed cost and selling price no greater than per-item cost, there is no nonnegative break-even quantity.
Rule: A continuous model permits every nonnegative real input, including fractions between whole counts. A fractional break-even input is a crossing in that model. For whole items, there is no exact break-even count unless that input is whole; the next whole count is the first with nonnegative profit when p > v.
The equal height is the shared dollar total.
Reminder
System of equations. Cost equals revenue uses the same method as setting two line outputs equal.
Signed subtraction. 20 − 22 = −2 records a loss of 2.
The same idea, five ways
Say it
Say: money in minus money spent tells what remains.
Write it
Break-even has no profit and no loss because cost equals revenue.
In math
P(x) = R(x) − C(x)
P(x) = 0
R(x) > C(x): profit
R(x) < C(x): loss
Like
You have paid back the table fee when each sale's contribution has added up to it.
See itAt five items, both lines have height $25.
Worked exampleA small break-even point
You need the number of $5 sales that cover a $10 setup fee and $3 production cost per item. Then repeat the interpretation if the setup fee is $5 instead of $10.
The quantity coordinate is 5; the dollar coordinate is 25.The model crosses between 2 and 3 whole sales; 3 is the first count whose revenue covers cost.
Set 10 + 3x = 5x.The same quantity must make cost and revenue equal.
Subtract 3x: 10 = 2x. Divide by 2 to find x = 5.Each sale contributes $2 toward the fixed fee.
Plug in: C(5) = 10 + 15 = 25 and R(5) = 25.Both original dollar amounts must match at the found quantity.
Compute P(5) = 25 − 25 = 0.The equality means no money remains after expenses.
For a $5 setup fee, set 5 + 3x = 5x. Subtract 3x to get 5 = 2x and divide by 2: x = 52 = 2.5.Equality finds the continuous-model crossing, even when it falls between actual whole-item counts.
Plug in 2.5: C(2.5) = 5 + 7.5 = 12.5 and R(2.5) = 12.5.Substitution confirms the model's fractional crossing; 2.5 is exact, not rounded.
At 2 whole items, profit is 10 − 11 = −1. At 3, profit is 15 − 14 = 1.Only whole sales counts are allowed. Two sales still lose money; the next whole count, 3, covers the expense.
Answer
With the $10 setup fee: 5 items.
Cost = revenue = $25.
Profit = $0.
With a $5 setup fee: the model crosses at 2.5 items.
Model cost = revenue = $12.50.
There is no exact whole-item break-even.
Three sales first give nonnegative profit, namely $1.
Check Five contributions of $2 total $10, exactly the setup fee.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: the break-even answer is $25 with no quantity.
The dollar coordinate tells the money involved, while the question asks how many sales are needed.
✓ Instead: Report 5 items, where both amounts are $25.
Tips and tricks
Tip: Label the two coordinates with units so you do not exchange quantity and dollars.
Strategy: step by step
1. Define the input quantity and its units; count sold items as nonnegative whole numbers.
2. Put fixed expense at the cost intercept and per-item expense at the cost slope.
3. Put selling price at the revenue slope, with revenue 0 before any sales.
4. Set cost equal to revenue and solve for the quantity.
5. Plug the quantity into both functions, then report quantity and dollars together.
6. Compare revenue with cost to tell profit from loss.
7. If the calculated quantity is between whole items, the model has no exact whole-item break-even. When each sale contributes a positive amount, use the next whole count to reach nonnegative profit.
Strategy
Strategy: build and solve a break-even model
1
Are the cost and revenue formulas already supplied?
YesUse them directly and equate their outputs.
NoBuild them from the fixed fee, production rate and selling price.
↓
2
With positive fixed cost, is p > v?
YesThe contribution p − v is positive; divide F by it.
NoNo nonnegative quantity covers the positive fixed expense under this model.
↓
3
Is the solved quantity a whole number when whole items are required?
YesIt is an exact break-even count; report its quantity and dollar amount.
NoThere is no exact whole-item break-even. With p > v, the next whole count is the first with nonnegative profit; check that count in both formulas.
Identify fixed cost F, per-item cost v and sale price p.
Write C(x) = F + vx and R(x) = px.
Set F + vx = px, then solve for x.
Plug the found quantity into both functions.
Report quantity, common dollar amount and zero profit.
If whole items are required, distinguish an exact whole crossing from a fractional model crossing.
Worked exampleThe draft's break-even model
You need the quantity at which C(x) = 0.01x + 25,000 and R(x) = 0.02x have the same dollar amount.
The shared point is where the cost and revenue heights are equal.
Set 0.01x + 25,000 = 0.02x.The shared quantity must give equal cost and revenue.
Subtract 0.01x from both sides: 25,000 = 0.01x.This isolates the one-cent-per-unit contribution available to pay the fixed expense.
Divide by 0.01: x = 2,500,000.Dividing by one hundredth is multiplying by 100, so this finds the number of units needed.
Plug into cost: C(2,500,000) = 0.01 × 2,500,000 + 25,000 = 25,000 + 25,000 = 50,000.The found input must reproduce the original total expense.
Plug into revenue: R(2,500,000) = 0.02 × 2,500,000 = 50,000.The same input must give the same sales total.
Profit is 50,000 − 50,000 = 0.Equal dollars mean all revenue pays cost.
Answer
2,500,000 units.
Cost = revenue = $50,000.
Break-even point: (2,500,000, 50,000).
Check Each unit contributes $0.02 − $0.01 = $0.01. Multiplying 2,500,000 by $0.01 gives $25,000, exactly the fixed cost.
Ladder: from easy to exam-hard. Press Try it first on any rung to hide its steps and use them as hints.
Rung 1Rung 1: a small setup bill
You need to find how many sales cover a $6 setup fee when each item costs $2 to make and sells for $5. Assume each item made is sold.
At two sales, both the expense and sales heights are $10.
Write C(x) = 6 + 2x and R(x) = 5x.The one-time bill starts the cost; repeated item costs and sale prices multiply the quantity.
Set 6 + 2x = 5x.This selects a quantity at which money spent equals money received.
Subtract 2x from both sides: 6 = 3x.Each sale contributes 5 − 2 = 3 dollars toward the setup bill.
Divide by 3: x = 2.This finds how many $3 contributions cover the $6 fixed expense.
Check C(2) = 6 + 4 = 10 and R(2) = 5 × 2 = 10.Both original rules must return the same dollar amount.
Answer
2 items.
Cost = revenue = $10.
Break-even point: (2, 10).
Profit: $0.
Check Two contributions of $3 total $6, independently covering the setup bill.
Rung 2Rung 2: prices with decimals
You need the break-even quantity with an $84 setup fee, production cost $1.50 per item, and sale price $3.25 per item. Assume every item produced is sold.
The cost begins at $84; the revenue catches it at 48 items.
Write C(x) = 84 + 1.50x and R(x) = 3.25x. Set 84 + 1.50x = 3.25x.A common input must give equal costs and revenue at break-even.
Multiply every term by 100: 8,400 + 150x = 325x.This clears the two-place decimal coefficients and scales the fixed fee too.
Subtract 150x: 8,400 = 175x.The remaining coefficient corresponds to each sale's $1.75 contribution, measured in cents.
Divide by 175: x = 48, because 175 × 48 = 8,400.This finds the quantity paying the setup bill.
Check C(48) = 84 + 1.50 × 48 = 84 + 72 = 156.The original cost formula verifies the total expense.
Check R(48) = 3.25 × 48 = 156.The original revenue formula gives the same total, so the solved count works.
Answer
48 items.
Cost = revenue = $156.
Break-even point: (48, 156).
Check The contribution route gives 48 × $1.75 = $84, exactly the starting bill. All decimals here are exact prices.
Rung 3Rung 3: a fraction is not a sale count
You need the model crossing and the first whole-item count covering costs when setup is $19, production is $4 per item, and sale price is $7. All produced items are sold.
The lines meet between 6 and 7 items, while seven whole sales cover the expense.
Write C(x) = 19 + 4x and R(x) = 7x, then set 19 + 4x = 7x.The equality selects equal dollar amounts at the same quantity.
Subtract 4x: 19 = 3x. Divide by 3: x = 193 = 6 + 13.Each sale contributes $3, and six contributions do not yet cover $19.
At the model input 193, C = 19 + 763 = 573 + 763 = 1333, and R = 1333.Equal thirds verify the continuous-model crossing exactly, without rounding.
At 6 items, C(6) = 19 + 24 = 43 and R(6) = 42. Profit is 42 − 43 = −1.The lower whole count has not covered costs.
At 7 items, C(7) = 19 + 28 = 47 and R(7) = 49. Profit is 49 − 47 = 2.The next whole count covers costs because each extra sale adds a positive $3 contribution.
Answer
Model crossing: (193, 1333).
No exact whole-item break-even exists.
First whole count covering costs: 7 items.
At 7 items: cost $47.
At 7 items: revenue $49.
At 7 items: profit $2.
Check Profit at whole count x is 3x − 19. It is −1 at 6 and 2 at 7, so the first acceptable count is 7, agreeing with both cost/revenue substitutions.
Rung 4Rung 4: sales that cannot cover the bill
You need to decide whether a $21 setup fee can be covered if each item costs $4 to make. Check a $4 sale price, then a $3 sale price. Assume all produced items are sold and x is nonnegative.
With equal price and per-item cost, revenue always remains $21 below cost.When price is below per-item cost, the loss grows with every allowed sale.
With a $4 sale price, C(x) = 21 + 4x and R(x) = 4x. Set 21 + 4x = 4x.Break-even still asks for equal dollar outputs.
Subtract 4x from both sides: 21 = 0. There is no solution.The contribution is $4 − $4 = $0, so no sale pays any part of the positive setup bill.
Check x = 2: cost is 21 + 8 = 29, revenue is 8, and profit is −21.The original formulas show the same unpaid setup bill; it remains at every quantity.
With a $3 sale price, use R(x) = 3x. Set 21 + 4x = 3x, then subtract 4x: 21 = −x.Each sale now costs $1 more than it brings in.
Divide by −1: x = −21. Reject this input because item quantity must be 0 or greater.The algebraic crossing lies outside the context's domain, so it is not a break-even sales count.
Plug −21 into both algebraic formulas: C(−21) = 21 − 84 = −63 and R(−21) = −63. Then check an allowed quantity: C(2) = 29 and R(2) = 6.The first check verifies the algebra but does not make negative items meaningful. The allowed count still loses money.
For every allowed x, profit is 3x − (21 + 4x) = −21 − x < 0.The initial loss of $21 grows as nonnegative quantity grows; no allowed input breaks even.
Answer
Sale price $4: no break-even quantity.
Sale price $3: no nonnegative break-even quantity.
The algebraic crossing at x = −21 is outside the allowed sales domain.
Check The contributions are 0 and −1 dollar per item. Neither can pay a positive $21 setup bill, confirming both results directly.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: $50,000 at the crossing is the company's profit.
The company also spent $50,000, so none of the sales money is left after cost.
✓ Instead: Profit is $50,000 − $50,000 = $0.
✗ Not this: Counterexample: two telephone plans with the same per-minute rate but different flat fees must eventually cost the same.
Their slopes match and their intercepts differ, so the fee gap stays fixed at every usage amount.
✓ Instead: They are parallel cost lines and do not have an equal-cost quantity.
Tips and tricks
Tip: Memory cue: break-even means the difference breaks even at 0.
Tip: Before solving, say what x counts and what each output measures.
Understand, then rebuild it when needed: the formula F ÷ (p − v) comes from cost equals revenue; the reason matters more than memorizing the division.
Know cold Coordinate order: across, then up or down; x comes before y. Slope: rise over run. Axis crossings: the other coordinate is 0. HOY: Horizontal, 0 slope, Y equation. VUX: Vertical, Undefined slope, X equation. Parallel: match slopes, compare intercepts. Perpendicular: flip and sign, then check product −1, except horizontal against vertical. System solutions: shared input, shared output.
Understand, then rebuild it when needed Tables come from substituting inputs, so you do not need to memorize individual coordinates. Transformation order comes from multiplication before addition. An x-intercept comes from solving output = 0. A line through a point comes from measuring changes from that point. An intersection comes from equal outputs. Break-even comes from cost equals revenue.
Put on the cheat sheet y = mx + b. m = y2 − y1x2 − x1, x2 ≠ x1. y − y1 = m(x − x1). b = y1 − mx1. x-intercept: x = −bm, m ≠ 0. Perpendicular slope: −1m, m ≠ 0 and slope finite. C(x) = F + vx; R(x) = px; profit = R(x) − C(x). Break-even: C(x) = R(x). These are also the compact printable formulas for this section. For the closed-book exam, practice rebuilding each from its reason.
Trap. Confusing the shared revenue with profit. At break-even both cost and revenue can be large, but profit is their difference, which is 0.
