Translating Words to Expressions

Write an algebraic expression for: "five more than a number"

Choose a variable for the unknown number: Let $n$ represent the unknown number = Variable: $n$

Identify the operation: "More than" means addition = Operation: $+$

Write the expression: Five more than $n$ means $n + 5$ = $n + 5$

Answer: $n + 5$ (or equivalently $5 + n$)

Evaluating an Expression

Evaluate $3x + 7$ when $x = 4$

Write the original expression: $3x + 7$ = Expression ready

Substitute the value: Replace $x$ with $4$: $3(4) + 7$ = $3(4) + 7$

Multiply first: $3 \times 4 = 12$ = $12 + 7$

Add: $12 + 7 = 19$ = $19$

Answer: $19$

Multi-Variable Expression

Evaluate $2a + 3b$ when $a = 5$ and $b = 2$

Write the original expression: $2a + 3b$ = Expression ready

Substitute both values: Replace $a$ with $5$ and $b$ with $2$: $2(5) + 3(2)$ = $2(5) + 3(2)$

Multiply each term: $2 \times 5 = 10$ and $3 \times 2 = 6$ = $10 + 6$

Add the results: $10 + 6 = 16$ = $16$

Answer: $16$

Real-World Expression

A streaming service costs 15 dollars per month plus 3 dollars for each extra user. Write an expression for the total monthly cost with $u$ extra users, then find the cost for 4 extra users.

Identify the fixed cost: Base cost is 15 dollars per month = Fixed: $15$

Identify the variable cost: Each extra user costs 3 dollars, so $u$ users cost $3u$ dollars = Variable: $3u$

Write the expression: Total cost = fixed + variable = $15 + 3u$

Substitute $u = 4$: $15 + 3(4) = 15 + 12 = 27$ = 27 dollars

Answer: Expression: $15 + 3u$; Cost for 4 extra users: 27 dollars

Mistake: Writing $3x$ as $3 + x$ instead of $3 \times x$

Why: In algebra, when a number is written next to a variable without a sign, it means multiplication. $3x$ means "3 times $x$", not "3 plus $x$".

Correct: $3x = 3 \times x$. So if $x = 5$, then $3x = 15$ (not $8$).

Mistake: Forgetting order of operations when evaluating

Why: When substituting values, you must still follow PEMDAS/BODMAS. Multiplication comes before addition.

Correct: For $2 + 3x$ with $x = 4$: First $3 \times 4 = 12$, then $2 + 12 = 14$.

Mistake: Using the same variable for different unknowns

Why: If you have two different unknown quantities, they need different variables.

Correct: If age and height are both unknown, use $a$ for age and $h$ for height, not $x$ for both.

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.

If shirts are 25 dollars each: $40 + 2(25) = 40 + 50 = 90$ dollars total.

Distance and Travel

If you drive at $s$ kilometers per hour for $t$ hours, you travel $s \times t$ kilometers.

Driving at 60 km/h for 3 hours: $60 \times 3 = 180$ km.

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.

Saving 15€ per week: After 1 week: $50 + 15 = 65€$.

Q: Why do we use letters instead of just leaving blanks?

A: Letters are clearer and allow us to write rules. For example, $A = lw$ tells us exactly how to find area, while "Area = ___ times ___" is ambiguous.

Q: Does it matter which letter I use for a variable?

A: Usually no, but some letters have common meanings: $x$ for unknowns, $t$ for time, $d$ for distance, $n$ for number of items. Choose letters that make sense for your problem.

Q: What is the difference between an expression and an equation?

A: An expression is a mathematical phrase without an equals sign (like $3x + 5$). An equation has an equals sign and shows two things are equal (like $3x + 5 = 14$).