Variables and Algebraic Expressions
Learn what variables are and how to write, read, and evaluate algebraic expressions.
Definition
- Numbers (constants like , , )
- Variables (letters like , , )
- Operations (like , , , )
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Worked Examples
Write an algebraic expression for: "five more than a number"
Choose a variable for the unknown number
Let represent the unknown number → Variable:
Identify the operation
"More than" means addition → Operation:
Write the expression
Five more than means →
Answer: (or equivalently )
Common Mistakes
Writing as instead of
Why it's wrong: In algebra, when a number is written next to a variable without a sign, it means multiplication. means "3 times ", not "3 plus ".
Correct: . So if , then (not ).
Forgetting order of operations when evaluating
Why it's wrong: When substituting values, you must still follow PEMDAS/BODMAS. Multiplication comes before addition.
Correct: For with : First , then .
Using the same variable for different unknowns
Why it's wrong: If you have two different unknown quantities, they need different variables.
Correct: If age and height are both unknown, use for age and for height, not for both.
Interactive Visual
Balance Scale
Solution: x = 4
Interactive Sandbox
Expression Calculator
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Practice Problems
15 problemsWhich of the following is a variable?
Why It Matters
- Formulas: The area of a rectangle is (length times width)
- Real life: If a movie ticket costs 12 dollars, the cost for people is dollars
- Science: Distance equals speed times time:
- Computer programming: Variables store changing values in every program
Real World Applications
Shopping and Discounts
Stores use expressions to calculate prices. If jeans cost 40 dollars and shirts cost $s$ dollars each, buying 2 shirts with jeans costs $40 + 2s$ dollars.
Example:
If shirts are 25 dollars each: dollars total.
A pizza shop charges 12 dollars for a pizza plus 2 dollars per topping.
Write an expression for the cost with toppings, then find the cost of a pizza with 5 toppings.
Step 1: Write the mathematical expression
Cost = base price + (price per topping × number of toppings):
Distance and Travel
If you drive at $s$ kilometers per hour for $t$ hours, you travel $s \times t$ kilometers.
Example:
Driving at 60 km/h for 3 hours: km.
A train travels at 80 kilometers per hour.
Write an expression for the distance after hours, then find how far it goes in 4 hours.
Step 1: Write the mathematical expression
Distance = speed × time:
Savings and Money
Expressions help track money over time. If you have 50 euros and save $w$ euros each week, after one week you have $50 + w$ euros.
Example:
Saving 15€ per week: After 1 week: .
You have 100€ and spend 8€ each day on lunch.
Write an expression for the money left after days.
Step 1: Write the mathematical expression
Money left = starting amount - (daily spending × days):
Key Takeaways
- 1A variable is a letter representing an unknown or changing number
- 2An algebraic expression combines numbers, variables, and operations (no equals sign)
- 3When a number is next to a variable (like ), it means multiplication:
- 4To evaluate an expression, substitute values for variables and calculate
- 5Always follow order of operations (PEMDAS) when evaluating
Frequently Asked Questions
Glossary
- Variable
- A letter that represents an unknown or changing number (e.g., , , )
- Algebraic expression
- A combination of numbers, variables, and operations without an equals sign
- Constant
- A number that does not change (e.g., , , )
- Coefficient
- The number multiplied by a variable (in , the coefficient is )
- Term
- A single part of an expression separated by or (in , the terms are and )
- Evaluate
- To find the value of an expression by substituting numbers for variables