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Mathematical
Modeling

Learn to translate real-world situations into equations, tables, and graphs, then interpret your results to make meaningful predictions.

Pre-Algebra

What Is Mathematical Modeling?

A mathematical model is a representation of a real-world situation using mathematical language: equations, tables, graphs, or expressions. It lets us organize information, find patterns, and make predictions.

Why Modeling Matters

Predict
Forecast future outcomes: how much money you will save, how far you will travel.
Decide
Compare options: which phone plan is cheaper after 6 months?
Explain
Communicate patterns clearly: show why a trend is increasing or decreasing.

Key Vocabulary

1
Variable: A letter (like \( x \) or \( n \)) that stands for an unknown or changing quantity.
2
Expression: A math phrase with numbers, variables, and operations, such as \( 3x + 5 \).
3
Equation: A statement that two expressions are equal: \( y = 3x + 5 \).
4
Interpret: Explain what the mathematical answer means back in the real-world context.

The Modeling Cycle

1. Identify
2. Formulate
3. Compute
4. Interpret
5. Validate
revise?

Building an Equation from Words

The foundation of mathematical modeling is converting a situation into an equation. The general form for many real-world linear situations is:

Your job is to identify what each part represents in the story, then substitute the numbers from the problem.

Expression vs Equation vs Model

Expression Equation Model
Example \( 3x + 5 \) \( y = 3x + 5 \) \( C = 3n + 5 \)
Has \( = \) sign? No Yes Yes
Context? No Not necessarily Yes, always
Purpose Calculate a value Show a relationship Represent a situation

How to Build a Model

1
Read carefully. Underline what changes (variable) and what stays fixed (constant).
2
Define variables. Choose letters and write what each represents, with units.
3
Write the equation. Use the pattern \( y = mx + b \) or another relationship that fits.
4
Solve. Substitute known values and compute.
5
Interpret. State the answer in a full sentence using the context and units.

Example 1: Lemonade Stand

Maya spends $10 on supplies and sells each cup for $2. Write a model for her profit \( P \) after selling \( c \) cups.

Cups sold (\( c \))
Profit ($)
-10
-5
0
5
10
1
3
5
7
9
(0, -10)
(5, 0) break-even
(10, 10)

Interpreting the Lemonade Model

Maya wants to know: How much profit after selling 15 cups?

1
Substitute: \( P = 2(15) - 10 \)
2
Compute: \( P = 30 - 10 = 20 \)
3
Interpret: After selling 15 cups, Maya earns a profit of $20.

Example 2: Comparing Plans

Two streaming services:
Plan A: $5 per month, no signup fee.
Plan B: $3 per month, plus a $12 signup fee.
Which is cheaper after \( m \) months?

Months (\( m \))
Cost ($)
0
10
20
30
40
50
1
3
5
7
9
Plan A
Plan B
(6, $30)

Finding the Break-Even Point

When are the two plans equal? Set the models equal and solve:

1
Set equal: \( 5m = 3m + 12 \)
2
Subtract \( 3m \): \( 2m = 12 \)
3
Divide by 2: \( m = 6 \)
4
Interpret: Both plans cost $30 at 6 months. If you plan to subscribe for more than 6 months, choose Plan B.

Example 3: Table to Equation

A taxi charges a flat fee plus a per-mile rate. Use the table to write a model.

Miles (\( d \)) 0 2 4 6 8
Cost (\( C \)) $3 $8 $13 $18 $23

Not All Models Are Linear

Some real-world situations grow faster and faster. When a quantity is multiplied by the same factor at regular intervals, we use an exponential model:

Days (\(d\))
Bacteria (\(B\))
0
50
100
150
200
0
1
2
3
4
5
Linear
Exponential

Common Mistakes

✕ Forgetting the Starting Value
Writing \( C = 2.5d \) instead of \( C = 2.5d + 3 \). Always check: what happens when the variable is 0?
✕ Labeling Without Context
Saying "the answer is 20" instead of "Maya earns $20 profit." Always include units and meaning.
✕ Confusing Rate with Total
Using the total cost as the rate. If 4 miles costs $13, the rate is NOT $13, it is \( \frac{13 - 3}{4} = \$2.50 \) per mile.

Modeling Cheat Sheet

Step What to Do Example
1. Identify Find what changes and what is fixed Cups change, supply cost is fixed
2. Define Assign variables with units \( c \) = cups, \( P \) = profit ($)
3. Write Form the equation \( P = 2c - 10 \)
4. Solve Substitute and compute \( P = 2(15) - 10 = 20 \)
5. Interpret Answer in a full sentence "Maya earns $20 profit."

Question 1 of 5

A gym charges a $25 signup fee and $15 per month. Which equation models the total cost \( C \) after \( m \) months?

Question 2 of 5

Using the model \( C = 15m + 25 \), what is the total cost after 8 months?

Question 3 of 5

A table shows: when \( x = 0 \), \( y = 7 \); when \( x = 3 \), \( y = 19 \). What is the rate of change?

Question 4 of 5

A car rental costs $40 per day plus $0.25 per mile. You rent for 3 days and drive 200 miles. What interpretation is correct?

Question 5 of 5

In the equation \( y = 4x + 7 \), what does the 7 represent in a real-world context?