Chapter 23

Linear Equations, Graphs, and Business Relationships

Translate a situation, read a line, and locate break-even

A delivery fee, an hourly paycheck, and a food stand's costs can all become equations. Learn to solve those equations, graph their relationships, and explain what their answers mean. All businesses, prices, and earnings in this chapter are hypothetical; round money to the nearest cent when needed.

23.1

Writing and Solving Linear Equations

Learning objectives

After this section you will be able to:

Translate fixed charges and per-unit charges into equations.
Solve linear equations using inverse operations.
Check solutions and interpret their units.

Real-life situation

A neighborhood equipment rental

A hypothetical Philadelphia neighborhood group rents equipment for a cleanup. The charge is $18 plus $14 per hour. An equation lets the group work backward from its bill to the rental time.

Hours h	Charge ($)
0	18
2	46
5	88

Figure 23.1 Hypothetical rental charges

The concept

A variable stands for an unknown or changing quantity. Before writing an equation, define the variable and its units. If h is rental time in hours, 14h means 14 dollars per hour multiplied by h hours.

An equation states that two expressions have equal values. A fixed charge appears once, while a per-unit charge is multiplied by the number of units. The rental model is C = 18 + 14h; when the bill is $88, it becomes 88 = 18 + 14h.

Solve by undoing operations in reverse order. Subtract an added amount before dividing by a coefficient. Perform the same operation on both sides so that the equality remains true. When parentheses occur, distribute correctly or undo the outside operation first.

Finally, substitute the proposed answer into the original equation. A correct calculation must also fit the situation: negative rental time is not meaningful, and a business that bills only whole hours may require a whole-number interpretation.

Writing and Solving Linear Equations: solution map

1
Identify the question
The rental bill is $88. At $18 plus $14 per hour, how many hours were charged?
2
Calculation 1
Define h as the number of rental hours.
3
Calculation 2
Write 18 + 14h = 88.
4
Calculation 3
Subtract 18: 14h = 70.
5
Calculation 4
Divide by 14: h = 5.
6
Calculation 5
Check: 18 + 14 × 5 = 18 + 70 = 88.
7
Interpret and check
The bill covers 5 hours.

Figure 23.1.V Writing and Solving Linear Equations: solution map. Hypothetical teaching data.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

variable cost
A cost that rises or falls with the number of items made or sold.
fixed cost
A cost that stays the same no matter how many items are sold, like rent.
revenue
The total money a business brings in from sales, before any costs are paid.

Key terms

Variable: A symbol representing an unknown or changing quantity.
Coefficient: A number multiplying a variable.
Constant: A fixed number in an expression.
Linear equation: An equation that can be written ax + b = c, with a nonzero.
Inverse operations: Operations that undo each other, such as addition and subtraction.

Formulas & rules

Fixed-plus-variable charge
C = F + r × x
F is the fixed charge and r is the charge per unit.
Solve a linear equation
x = (c − b) ÷ a
For ax + b = c, with a ≠ 0.
Distributive property
a × (x + b) = ax + ab

Worked examples

Example 1Work backward from a rental bill

The rental bill is $88. At $18 plus $14 per hour, how many hours were charged?

Solution

1Define h as the number of rental hours.
2Write 18 + 14h = 88.
3Subtract 18: 14h = 70.
4Divide by 14: h = 5.
5Check: 18 + 14 × 5 = 18 + 70 = 88.

Answer: The bill covers 5 hours.

Example 2Undo grouped operations

Three identical supply kits cost $90 altogether. Each kit costs 4x + 6 dollars. Find x.

Solution

1Write 3(4x + 6) = 90.
2Divide both sides by 3: 4x + 6 = 30.
3Subtract 6: 4x = 24.
4Divide by 4: x = 6.
5Check: 3(4 × 6 + 6) = 3 × 30 = 90.

Answer: x = 6; each kit costs $30.00.

Practice exercises 23.1

Evaluate C = 12h + 15 for h = 6.
Solve 10x + 15 = 95.
Evaluate 3(x + 4) for x = 7.
Solve 4(5x + 3) = 72.
A student says 2x + 6 = 22 gives x = 14. Explain and correct the error.

Section 23.1 recap

Define the variable, write the equation, and undo operations on both sides.
Check the result in the original equation and explain its units.

Answer key with explanations: Section 23.1

Practice exercises

Substitute 6: C = 12 × 6 + 15 = 72 + 15 = $87.00.
Subtract 15 to get 10x = 80; divide by 10 to get x = 8. Check: 80 + 15 = 95.
First add inside parentheses: 7 + 4 = 11. Then 3 × 11 = 33.
Divide by 4: 5x + 3 = 18. Subtract 3: 5x = 15. Divide by 5: x = 3.
Subtracting 6 gives 2x = 16, not x = 16; dividing by 2 gives x = 8. Check: 2 × 8 + 6 = 22.

Recap quiz

5: 8x = 40, so x = 40 ÷ 8 = 5.
4x + 8: Distribute 4 to both terms: 4x + 4 × 2 = 4x + 8.
$44: 9 + 7 × 5 = 9 + 35 = $44.00.

23.2

Slope, Intercepts, and Graphing Lines

Learning objectives

After this section you will be able to:

Calculate slope from two ordered pairs.
Interpret slope and the vertical intercept in context.
Plot a linear relationship with labeled axes and an appropriate domain.

Real-life situation

Preparing a delivery-cost graph

A hypothetical local delivery service charges $30 plus $5 per mile. Its table supplies points for a graph: miles belong on the horizontal axis, and total dollars belong on the vertical axis.

Miles x	Total cost C ($)	Ordered pair
0	30	(0, 30)
2	40	(2, 40)
6	60	(6, 60)

Figure 23.2 Plotting data for C = 5x + 30

The concept

A coordinate graph pairs each input with an output. The horizontal axis usually shows the input x, and the vertical axis shows the output y. Label both axes with quantities and units, and choose evenly spaced numerical scales.

Slope measures the change in output divided by the change in input. Subtract coordinates in the same order in the numerator and denominator. For a delivery model, slope has units of dollars per mile, not simply dollars.

In y = mx + b, m is the slope and b is the vertical intercept, the output when x = 0. Positive slope means the output rises as x increases; negative slope means it falls. A horizontal line has slope zero. A vertical line has undefined slope because its change in x is zero.

To graph a line, plot the intercept and at least one additional point, or calculate a small table. Draw the line through the points, but restrict it to meaningful inputs. Distance may vary continuously, whereas counts of tickets or shirts use separate whole-number points.

Slope, Intercepts, and Graphing Lines: solution map

1
Identify the question
Use (2, 40) and (6, 60) to find the delivery equation and predict the cost for 8 miles.
2
Calculation 1
Slope: m = (60 − 40) ÷ (6 − 2) = 20 ÷ 4 = 5.
3
Calculation 2
Use (2, 40): b = 40 − 5 × 2 = 30.
4
Calculation 3
Write C = 5x + 30.
5
Calculation 4
Plot (0, 30), (2, 40), and (6, 60) on axes labeled miles and dollars; draw the line for x ≥ 0.
6
Calculation 5
At 8 miles: C = 5 × 8 + 30 = 70.
7
Interpret and check
Slope $5 per mile, intercept $30.00, and an 8-mile charge of $70.00.

Figure 23.2.V Slope, Intercepts, and Graphing Lines: solution map. Hypothetical teaching data.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

profit
What is left when expenses are subtracted from revenue.
break-even point
The sales level where revenue exactly equals total cost, so there is no profit and no loss.
monthly payment
The amount paid each month toward a loan or installment plan.

Key terms

Ordered pair: A point written (x, y), with horizontal coordinate first.
Slope: Change in vertical coordinate divided by change in horizontal coordinate.
Vertical intercept: Where a graph crosses the vertical axis; x = 0.
Domain: The allowed input values of a relationship.
Linear relationship: A relationship with a constant rate of change whose graph lies on a straight line.

Formulas & rules

Slope
m = (y₂ − y₁) ÷ (x₂ − x₁)
The two x-values must differ.
Slope-intercept form
y = mx + b
Find the intercept
b = y − mx

Worked examples

Example 1Graph a delivery charge

Use (2, 40) and (6, 60) to find the delivery equation and predict the cost for 8 miles.

Solution

1Slope: m = (60 − 40) ÷ (6 − 2) = 20 ÷ 4 = 5.
2Use (2, 40): b = 40 − 5 × 2 = 30.
3Write C = 5x + 30.
4Plot (0, 30), (2, 40), and (6, 60) on axes labeled miles and dollars; draw the line for x ≥ 0.
5At 8 miles: C = 5 × 8 + 30 = 70.

Answer: Slope $5 per mile, intercept $30.00, and an 8-mile charge of $70.00.

Example 2A decreasing balance

A prepaid balance is $26 at 0 sessions and $14 after 4 sessions. Find the equation and the balance after 6 sessions.

Solution

1Slope: m = (14 − 26) ÷ (4 − 0) = −12 ÷ 4 = −3.
2The starting balance is the intercept, b = 26.
3Write B = −3s + 26.
4Substitute s = 6: B = −18 + 26 = 8.
5Plot only whole-number session counts while the balance remains nonnegative.

Answer: B = −3s + 26; after 6 sessions, $8.00 remains.

Practice exercises 23.2

Find the line through (0, 10) and (6, 34).
Using y = 4x + 10, find the point with x = 3.
Find the slope through (0, 17) and (4, 5), and interpret its sign.
Find the slope through (2, 12) and (7, 12).
A shirt-sales graph uses x for shirts sold. Should you connect every point as a continuous business model?

Section 23.2 recap

Slope is a rate with units; the intercept is the output at zero input.
Label axes and graph only inputs that make sense in the situation.

Answer key with explanations: Section 23.2

Practice exercises

m = (34 − 10) ÷ 6 = 4. The intercept is 10, so y = 4x + 10.
y = 4 × 3 + 10 = 22, so the point is (3, 22).
m = (5 − 17) ÷ 4 = −3. The output decreases by 3 units for every 1-unit increase in input.
m = (12 − 12) ÷ (7 − 2) = 0 ÷ 5 = 0. The graph is horizontal.
No. Sales counts are 0, 1, 2, and so on; a value such as 2.5 shirts is not an allowed input. Plot whole-number points even if they lie on a line.

Recap quiz

17: 2 × 5 + 7 = 17.
4: (31 − 7) ÷ (6 − 0) = 24 ÷ 6 = 4.
The cost at x = 0: Setting x = 0 gives C = b, the vertical intercept.

23.3

Revenue, Cost, Profit, and Break-Even

Learning objectives

After this section you will be able to:

Build linear revenue, cost, and profit models.
Find and interpret a break-even quantity.
Distinguish a model's prediction from guaranteed sales.

Real-life situation

A school-market stand

A hypothetical student stand sells handmade items for $12 each. It pays $80 to set up and $8 per item sold. The table compares money coming in with modeled costs.

Items sold	Revenue ($)	Cost ($)	Profit ($)
0	0	80	-80
20	240	240	0
40	480	400	80

Figure 23.3 Hypothetical stand results

The concept

Revenue is money earned from sales before expenses. If every item sells for p dollars, revenue from q items is pq. This model assumes a constant selling price and that q items actually sell.

Total cost combines fixed cost, which does not change over the modeled production range, with variable cost per item. A simple model is C = F + vq. If items are produced but not sold, production quantity and sales quantity must be modeled separately.

Profit is revenue minus total cost. The difference p − v is the contribution margin per item: the amount available to cover fixed costs and then contribute to profit. It is not the same as final profit per item.

Break-even occurs where revenue equals cost and profit is zero. On a graph, it is the intersection of the revenue and cost lines. When p > v, quantities above break-even produce a modeled profit; if the calculated break-even quantity is not whole, round upward to find the first whole-unit quantity with nonnegative profit.

These equations do not guarantee customers or account automatically for waste, refunds, taxes, or capacity limits. State which costs are included and test whether the assumed selling price and per-item cost remain reasonable.

Revenue, Cost, Profit, and Break-Even: solution map

1
Identify the question
For the stand in Figure 23.3, find break-even sales and the revenue at that point.
2
Calculation 1
Write R = 12q and C = 80 + 8q.
3
Calculation 2
Set revenue equal to cost: 12q = 80 + 8q.
4
Calculation 3
Subtract 8q: 4q = 80.
5
Calculation 4
Divide by 4: q = 20.
6
Calculation 5
Check: R = 12 × 20 = 240 and C = 80 + 8 × 20 = 240.
7
Interpret and check
Break-even is 20 items, with revenue and cost both $240.00.

Figure 23.3.V Revenue, Cost, Profit, and Break-Even: solution map. Hypothetical teaching data.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

loan
Money borrowed that must be repaid, usually with interest.
deposit
Money placed into a bank account.
budget
A plan that shows expected income and how it will be spent.

Key terms

Revenue: Sales income before subtracting expenses.
Fixed cost: A cost unchanged by quantity within the modeled range.
Variable cost: A cost that changes with the number of units.
Profit: Revenue minus total cost; a negative result is a loss.
Contribution margin: Selling price minus variable cost per unit.
Break-even: The point at which revenue equals cost.

Formulas & rules

Revenue
R = p × q
Total cost
C = F + v × q
Profit
P = R − C = (p − v) × q − F
Break-even quantity
q = F ÷ (p − v)
Requires p > v for a positive break-even quantity when F > 0.

Worked examples

Example 1Locate break-even

For the stand in Figure 23.3, find break-even sales and the revenue at that point.

Solution

1Write R = 12q and C = 80 + 8q.
2Set revenue equal to cost: 12q = 80 + 8q.
3Subtract 8q: 4q = 80.
4Divide by 4: q = 20.
5Check: R = 12 × 20 = 240 and C = 80 + 8 × 20 = 240.

Answer: Break-even is 20 items, with revenue and cost both $240.00.

Example 2Calculate profit at a sales target

What profit does the stand model predict if it sells 35 items?

Solution

1Revenue: R = 12 × 35 = 420.
2Cost: C = 80 + 8 × 35 = 80 + 280 = 360.
3Profit: P = 420 − 360 = 60.
4Check with contribution margin: (12 − 8) × 35 − 80 = 140 − 80 = 60.

Answer: The modeled profit is $60.00, assuming all 35 items sell and the listed costs apply.

Practice exercises 23.3

Find the stand's profit at 40 items.
A stand has R = 9q and C = 30 + 6q. Find break-even.
For R = 9q and C = 30 + 6q, find profit at 25 items.
Fixed cost is $54, price is $15, and variable cost is $9 per item. Find break-even.
Find profit at 5 items if R = 10q and C = 70 + 6q.

Section 23.3 recap

Revenue is not profit: subtract both fixed and variable costs.
Break-even solves R = C; a sales forecast is not a promise.

Answer key with explanations: Section 23.3

Practice exercises

P = (12 − 8) × 40 − 80 = 160 − 80 = $80.00.
9q = 30 + 6q gives 3q = 30, so q = 10. Both revenue and cost equal $90.00.
R = 225 and C = 30 + 150 = 180. Profit = 225 − 180 = $45.00.
Contribution margin = 15 − 9 = 6. Break-even = 54 ÷ 6 = 9 items.
Revenue = 50; cost = 70 + 30 = 100. Profit = 50 − 100 = −$50.00, a $50.00 loss.

Recap quiz

$24: P = (13 − 7) × 12 − 48 = 72 − 48 = $24.00.
8 items: q = 48 ÷ (13 − 7) = 48 ÷ 6 = 8.
There is no break-even quantity in this model: The contribution margin is zero, so P = −F regardless of quantity.

Chapter 23 Organizer and Review

Chapter Organizer

Every section at a glance: key formulas, key terms, and the main idea.

23.1 Writing and Solving Linear Equations
Key formulas
- Fixed-plus-variable charge: C = F + r × x
- Solve a linear equation: x = (c − b) ÷ a
- Distributive property: a × (x + b) = ax + ab
Key terms
Variable, Coefficient, Constant, Linear equation, Inverse operations
Remember
Define the variable, write the equation, and undo operations on both sides.
23.2 Slope, Intercepts, and Graphing Lines
Key formulas
- Slope: m = (y₂ − y₁) ÷ (x₂ − x₁)
- Slope-intercept form: y = mx + b
- Find the intercept: b = y − mx
Key terms
Ordered pair, Slope, Vertical intercept, Domain, Linear relationship
Remember
Slope is a rate with units; the intercept is the output at zero input.
23.3 Revenue, Cost, Profit, and Break-Even
Key formulas
- Revenue: R = p × q
- Total cost: C = F + v × q
- Profit: P = R − C = (p − v) × q − F
- Break-even quantity: q = F ÷ (p − v)
Key terms
Revenue, Fixed cost, Variable cost, Profit, Contribution margin, Break-even
Remember
Revenue is not profit: subtract both fixed and variable costs.

Cumulative Review

Mixed practice from this chapter and the two chapters before it.

Section 23.1
A student says 2x + 6 = 22 gives x = 14. Explain and correct the error.
Show answer
Subtracting 6 gives 2x = 16, not x = 16; dividing by 2 gives x = 8. Check: 2 × 8 + 6 = 22.
Section 23.2
A shirt-sales graph uses x for shirts sold. Should you connect every point as a continuous business model?
Show answer
No. Sales counts are 0, 1, 2, and so on; a value such as 2.5 shirts is not an allowed input. Plot whole-number points even if they lie on a line.
Section 23.3
Find profit at 5 items if R = 10q and C = 70 + 6q.
Show answer
Revenue = 50; cost = 70 + 30 = 100. Profit = 50 − 100 = −$50.00, a $50.00 loss.
Review of Section 21.5
Go or No-Go? You buy 4 shares at $10.00 and sell at $10.50. The spread costs a combined $0.10 per share and the app charges a $2 fee on the sale. Find the net result and explain your decision.
Show answer
Gross 4 × $0.50 = $2. Costs 4 × $0.10 = $0.40 plus $2 = $2.40. Net $2 − $2.40 = −$0.40. A gain on the screen became a loss after costs; No-Go at this size.
Review of Section 22.5
Find wraps’ unit share and revenue share.
Show answer
Unit share = 80 ÷ 400 × 100% = 20.00%; revenue share = 480 ÷ 2,040 × 100% ≈ 23.53%.

All prices, stores, people, and accounts in this chapter are hypothetical and for teaching only. Product references are generic categories; no brand endorses or is endorsed by this textbook.

Chapter 22 Chapter 24