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=

Representing
Constraints

Learn to translate real-world limits and rules into equations, and explain what every variable and number means.

Pre-Algebra

What Is a Constraint?

A constraint is a rule or limit that restricts what values are possible. In everyday life, constraints are everywhere: a budget caps how much you can spend, a time limit caps how long you can work, and a recipe fixes the ratio of ingredients.

Why Use Equations?

Precision
An equation removes ambiguity. Everyone reads the same rule the same way.
Flexibility
Once a constraint is an equation, you can solve for any unknown in it.
Communication
Equations are a universal language: they work across subjects and contexts.

Parts of a Constraint Equation

\({\color{#e2e8f0}3}{\color{#818cf8}x} + {\color{#fbbf24}5} = {\color{#4ade80}20}\)
Coefficient (rate per item) Variable (unknown quantity) Constant (fixed amount) Total (constraint limit)

The Core Rule

A constraint situation can be represented as:

This pattern works whenever a quantity changes at a steady rate and may include a one-time addition or starting value. The equation is a mathematical sentence that says the left side and the right side must always be equal.

The Balance Model

\(3x + 5\)
 Left side
\(20\)
 Right side =

Equations vs. Expressions

Equation Expression
Has an = sign? Yes No
Example \(3x + 5 = 20\) \(3x + 5\)
Shows a constraint? Yes: two sides must balance No: just a calculation
Can you solve it? Yes: find the unknown Only simplify

Equations vs. Inequalities

Equation
Uses =. States that two sides are exactly equal. Example: \(x + 7 = 15\). There is often one solution.
Inequality
Uses <, >, \(\leq\), or \(\geq\). States a range of possible values. Example: \(x + 7 \leq 15\). There are many solutions.

How to Build a Constraint Equation

1
Identify the unknown. What quantity are you trying to find? Assign it a variable (e.g., \(x\)).
2
Find the rate. Is there a "per unit" amount? That becomes the coefficient multiplied by your variable.
3
Spot fixed amounts. Any one-time fee, starting value, or extra cost? That becomes a constant you add (or subtract).
4
Set it equal to the total. The constraint (budget, time limit, total distance) goes on the other side of the = sign.
5
Label every part. Explain what each number and variable represents in the real-world context.

Example 1: Movie Budget

You have $20 to spend at the movies. A ticket costs $8. You spend a total of \(s\) dollars on snacks. If you spend exactly $20, what is the value of \(s\)?

Ticket $8 + Snacks
\(s\)
 = Budget $20
\(8 + s = 20\)

Example 2: Fundraiser

A school club needs to raise $150. They already have $30 from donations. They sell cookies for $4 each. How many cookies \(c\) must they sell to reach their goal?

Donations $30 + Cookie Sales
\(4c\)
 = Goal $150
\(30 + 4c = 150\)

Example 3: Road Trip

A family is driving 300 miles to a national park. They have already driven 75 miles. They drive at a steady 50 miles per hour. How many hours \(h\) until they arrive?

75 mi
\(50h\) miles remaining
 Total: 300 miles

Explain Each Part

Writing the equation is only half the job. You also need to explain what each part means in the context of the problem. Here is a template:

1
Name the variable. "Let \(c\) = the number of cookies sold."
2
Explain the coefficient. "The 4 means each cookie earns $4."
3
Explain constants. "The 30 is the $30 already donated."
4
Explain the total. "The 150 is the fundraising goal."

Common Mistakes

✕ Forgetting the = sign
Writing \(4c + 30\) instead of \(4c + 30 = 150\). Without = you have an expression, not a constraint.
✕ Swapping variable and constant
Writing \(30c + 4 = 150\) when cookies cost $4 each and $30 is fixed. The coefficient must match the per-unit rate.
✕ Not explaining parts
Just writing the equation without saying what each number and variable means in the real-world situation.

Summary Checklist

Step What to Do Example
1. Variable Choose a letter for the unknown \(c\) = cookies
2. Coefficient Identify the per-unit rate 4 dollars each
3. Constant Find any fixed starting amount 30 dollars donated
4. Total What must everything add up to? 150 dollar goal
5. Equation Combine with = sign \(30 + 4c = 150\)
6. Explain Label every part in context "4 means $4 per cookie"

Question 1 of 5

A gym charges a $25 sign-up fee plus $10 per month. You have $85 to spend. Which equation represents this constraint, where \(m\) is the number of months?

Question 2 of 5

In the equation \(6p + 12 = 42\), a store sells pencils for $6 each and charges a $12 shipping fee with a $42 budget. What does the 12 represent?

Question 3 of 5

Maya has $50. She buys a book for $14 and wants to buy notebooks that cost $9 each. She wants to spend all her money. Which equation models this constraint?

Question 4 of 5

A rectangle has a perimeter of 36 cm. The length is 10 cm. If \(w\) represents the width, which equation represents this constraint?

Question 5 of 5

Which statement best explains what the coefficient represents in the equation \(3.50d + 5 = 26\), where \(d\) is the number of drinks purchased and 5 is a tip?