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\( x+5 \)
 $

Modeling with
Expressions

Learn to translate real-world situations into algebraic expressions and equations: interpret variables, units, and constraints.

Pre-Algebra

What Is an Expression?

An algebraic expression is a combination of numbers, variables, and operations that represents a quantity. Unlike an equation, it does not have an equals sign.

Expression
\( 3x + 7 \) : a phrase that describes a value
Equation
\( 3x + 7 = 22 \) : a statement that two values are equal

Variables, Constants, and Operations

Variable
A letter (like \( x \) or \( n \)) that stands for an unknown or changing quantity
Constant
A fixed number that does not change, like \( 5 \) or \( 12 \)
Coefficient
The number multiplied by a variable, like the 3 in \( 3x \)
\( \textcolor{#e2e8f0}{3}\textcolor{#818cf8}{x} \textcolor{#94a3b8}{+} \textcolor{#4ade80}{7} \)
coefficient variable constant

What Does "Modeling" Mean?

Modeling means turning a real-world situation into a mathematical expression or equation. You identify the unknown quantity, assign a variable, and write the relationship using math.

Real-World Situation Identify the Variable Write the Expression

Translating Words into Math

Certain English words map to specific math operations. Recognizing these keywords is the first step to writing expressions.

English Phrase Operation Symbol
sum of, more than, increased byAddition\( + \)
difference, less than, decreased bySubtraction\( - \)
product of, times, eachMultiplication\( \times \) or \( \cdot \)
quotient of, divided by, perDivision\( \div \) or \( / \)
is, equals, results inEquals\( = \)

The Modeling Rule

Expression
\( 25m + 40 \)
 "total cost for m months" No equals sign Describes a quantity → Equation
\( 25m + 40 = 190 \)
 "cost equals $190" Has an equals sign Can be solved

Units and Constraints

Every variable and constant carries units (dollars, hours, miles). A constraint is a rule that limits what values a variable can take.

Units
If \( m \) = months and the rate is $25/month, then \( 25m \) gives an answer in dollars.
Constraints
Months must be a whole number \( \geq 0 \). You can't have \( m = -3 \) or \( m = 2.7 \) in this context.

Expression vs Equation

Feature Expression Equation
Equals sign?NoYes
PurposeDescribes a quantityStates two quantities are equal
Example\( 4n - 3 \)\( 4n - 3 = 17 \)
Can you "solve" it?Evaluate or simplifySolve for the variable
Real-world roleGeneral formulaFinding a specific answer

How to Model a Situation

1
Read carefully. Identify what changes (the unknown) and what stays fixed (constants).
2
Define the variable. Write "Let \( x \) = ..." with a clear description and unit.
3
Translate keywords. Match phrases to operations using the keyword table.
4
Write the expression or equation. Combine the variable, constants, and operations.
5
Check units and constraints. Do the units match? Do the values make sense in context?

Example 1: Movie Tickets

"A movie ticket costs $12. You also buy a popcorn for $6. Write an expression for the total cost of \( t \) tickets plus one popcorn."

1
Unknown: number of tickets → Let \( t \) = number of tickets
2
Constants: $12 per ticket, $6 for popcorn
3
Expression: \( 12t + 6 \)   (dollars)

Example 2: Saving Money

"Mia has $80 in her account. She saves $15 each week. Write an expression for her balance after \( w \) weeks, then write an equation to find when she reaches $200."

w = 0 $80 w = 1 $95 w = 2 $110 w = 3 $125 w = ? $200

Example 3: Perimeter of a Rectangle

"A rectangle's length is 3 more than twice its width. Write an expression for the perimeter."

\( w \)
\( w \)
\( 2w + 3 \)
\( 2w + 3 \)
1
Let \( w \) = width of the rectangle (in units)
2
Length = \( 2w + 3 \)
3
Perimeter = \( 2(2w+3) + 2w = 4w + 6 + 2w = 6w + 6 \)

Interpreting Each Part

Being able to read an expression and explain what each piece means in context is just as important as writing one.

\( \textcolor{#fbbf24}{0.08}\textcolor{#818cf8}{p} \textcolor{#94a3b8}{+} \textcolor{#4ade80}{5} \)
8% tax rate price (dollars) flat shipping fee Total cost = tax on price + shipping

Constraints in Context

Real-world problems always have constraints: limits on what the variable can be. Ignoring them leads to impossible answers.

People
Must be whole numbers \( \geq 0 \). You can't have 3.5 students.
Money
Usually \( \geq 0 \) (no negative balances unless debt is allowed).
Time
Always \( \geq 0 \). Negative time doesn't make physical sense.

Common Mistakes

✕ Reversed Subtraction
"5 less than \( x \)" is \( x - 5 \), not \( 5 - x \). The order matters in subtraction.
✕ Forgetting Units
Writing \( 12t + 6 \) without noting it's in dollars. Always label units.
✕ Ignoring Constraints
Accepting \( t = 3.7 \) tickets or \( n = -4 \) people. Check if answers are realistic.

Summary Checklist

Step Action Example
1. ReadFind the unknown\( t \) = tickets
2. DefineAssign variable + unitLet \( t \) = number of tickets
3. TranslateKeywords → operations"per" → multiply
4. WriteBuild expression/equation\( 12t + 6 \)
5. CheckVerify units + constraintsdollars; \( t \geq 1 \), integer

Question 1 of 5

A pizza costs $14 and each extra topping costs $2. Which expression represents the total cost for a pizza with \( t \) toppings?

Question 2 of 5

In the expression \( 5n + 20 \), where \( n \) = number of hours worked at a tutoring center, what does the 20 represent?

Question 3 of 5

"7 less than three times a number" translates to which expression?

Question 4 of 5

A car rental costs $35 per day plus a $50 insurance fee. You write the equation \( 35d + 50 = 225 \). What constraint must \( d \) satisfy?

Question 5 of 5

A store sells notebooks for $4 each and pens for $1 each. You buy \( n \) notebooks and \( p \) pens. Which expression gives the total cost in dollars?