Variables and Expressions

🔤 Variables and Expressions

Part 1 of 5 — The Anatomy of an Expression

Topics in This Part

🔑 Key Concept: A variable is a letter — like $x$, $n$, or $t$ — that holds a number's place. It may be a value we don't know yet, or one that's allowed to change. Algebra is just arithmetic with some of the numbers wearing letter-shaped masks.

A Variable Is a Stand-In for a Number

Imagine every movie ticket costs the same, but the price isn't posted. Call that price $p$. Now we can talk about it before we ever learn its value:

• $2p$ — the cost of 2 tickets
• $5p$ — the cost of 5 tickets
• $p + 3$ — "three dollars more than one ticket"

Variable vs. Constant

💡 Why use letters at all? A variable lets us write one rule that works for every number at once. The formula $2p$ gives the price of two tickets whether a ticket costs $8, $12, or $15 — we never have to rewrite it.

Concept Check 🎯

The Parts of an Expression

An expression combines numbers, variables, and operations — like $4x + 7$. It has no equals sign (an equals sign would make it an equation).

Every piece separated by a $+$ or $-$ is a term. Inside a term:

• the number multiplying the variable is the coefficient;
• a term that is only a number (no variable) is a constant term.

Dissecting $4x + 7$

⚠️ Watch out: If a variable appears with no number in front — like $x$ — its coefficient is $1$, not $0$. We simply don't bother writing the $1$. In the same way, $-x$ has coefficient $-1$.

Name That Part 🔽

Look at the expression $6y - 9$ and choose the correct description for each piece.

Expression or Equation? 🎯

Count the Terms 🧮

Terms are the pieces separated by $+$ or $-$. Count how many terms each expression has.

1) $5x + 2$ → how many terms? 2) $7a - 3b + 4$ → how many terms? 3) $9k$ → how many terms?

🔤 Variables and Expressions

Part 2 of 5 — Translating Words into Math

🔑 The Idea: Math is a language. To turn an English phrase into an expression, find the operation words and the quantities they connect, then build the expression piece by piece.

Operation Keywords

Certain words almost always signal a specific operation. Let "the number" be $n$.

⚠️ The "less than" trap: "$5$ less than a number" is $n - 5$, not $5 - n$. The phrase less than flips the order — you start with the number and take $5$ away. The same flip happens with "fewer than" and "subtracted from."

Concept Check 🎯

Worked Examples — Build It in Pieces

Translate the inner phrase first, then attach the outer operation.

Example 1 — "Twice a number, increased by 3."

• "twice a number" $= 2n$
• "increased by 3" adds $3$

$2n + 3$

Example 2 — "The quotient of a number and 5, decreased by 1."

• "quotient of a number and 5" $= \dfrac{n}{5}$
• "decreased by 1" subtracts $1$

$\dfrac{n}{5} - 1$

Example 3 — "8 less than three times a number."

• "three times a number" $= 3n$
• "8 less than" subtracts $8$ from it

$3n - 8$

💡 Reading left to right, "8 less than [three times a number]" pushes the $-8$ to the end, because less than flips the order.

Translate the Phrase 🔽

Match each English phrase to its expression. Let the number be $n$.

You Write the Coefficient 🧮

Each phrase translates to "(some number) $\cdot\, n$." Enter just that number — the coefficient.

1) "double a number" → $\;?\,n$ 2) "triple a number" → $\;?\,n$ 3) "a number times ten" → $\;?\,n$

🔤 Variables and Expressions

Part 3 of 5 — Evaluating by Substitution

🔑 The Idea: To evaluate an expression, substitute a number for each variable, then simplify using the order of operations. Replacing letters with numbers turns algebra back into arithmetic.

Substitute, Then Simplify

A variable written next to a number means multiply: $4x$ means $4 \cdot x$. So when you plug in a value, restore that multiplication with parentheses.

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

$4x + 7 = 4(3) + 7 = 12 + 7 = 19$

Example — Evaluate $2a - 5$ when $a = 6$

$2a - 5 = 2(6) - 5 = 12 - 5 = 7$

⚠️ Always use parentheses when you substitute. Writing $4(3)$ instead of $43$ keeps the multiplication visible and stops you from accidentally gluing the digits together.

Substitute and Solve 🧮

Evaluate each expression for the given value.

1) $5x + 1$ when $x = 4$ 2) $3n - 2$ when $n = 5$ 3) $\dfrac{m}{2} + 6$ when $m = 8$

Order of Operations (PEMDAS)

When an expression has several operations, follow this order:

Example — Evaluate $2x^2 + 1$ when $x = 3$

Handle the exponent first, then multiply, then add:

$2x^2 + 1 = 2(3)^2 + 1 = 2(9) + 1 = 18 + 1 = 19$

⚠️ Order matters! In $2x^2$, only the $x$ is squared — not the $2$. Square first ($3^2 = 9$), then multiply by $2$. Computing $(2 \cdot 3)^2 = 36$ is a classic mistake.

Concept Check 🎯

Expressions with Two Variables

Substitute each variable with its own value, then simplify.

Example — Evaluate $3a + 2b$ when $a = 4$ and $b = 5$

$3a + 2b = 3(4) + 2(5) = 12 + 10 = 22$

Example — Evaluate $ab - c$ when $a = 2,\; b = 6,\; c = 7$

$ab - c = (2)(6) - 7 = 12 - 7 = 5$

💡 Replace every copy of a variable with its number. If $a$ appears twice, plug the value into both spots.

Two Variables 🧮

Evaluate each expression using $a = 3$ and $b = 4$.

1) $2a + b$ 2) $ab$ 3) $5b - 3a$

🔤 Variables and Expressions

Part 4 of 5 — Simplifying: Like Terms & Distributing

🔑 The Idea: You can only add or subtract terms that share the exact same variable part — these are like terms. And the distributive property lets you clear parentheses. Together they shrink any expression to its simplest form.

What Are Like Terms?

Like terms have the same variable raised to the same power. Their coefficients can differ.

Combining Like Terms

Add or subtract the coefficients and keep the variable part unchanged:

$3x + 5x = (3 + 5)x = 8x$

⚠️ You do not touch the variable. $3x + 5x = 8x$, not $8x^2$. You're counting how many $x$'s you have — like $3$ apples $+\, 5$ apples $= 8$ apples, never "8 apples-squared."

Concept Check 🎯

Simplifying Longer Expressions

When an expression mixes different kinds of terms, group the like terms and combine each group separately. Constants combine with constants.

Example — Simplify $4x + 3 + 2x + 5$

$\underbrace{4x + 2x}_{6x} + \underbrace{3 + 5}_{8} = 6x + 8$

Example — Simplify $7a + 2b - 3a + 4b$

$\underbrace{7a - 3a}_{4a} + \underbrace{2b + 4b}_{6b} = 4a + 6b$

💡 Keep each term's sign with it. In $7a - 3a$, the $-3a$ travels with its minus sign. Reading a subtraction as "adding a negative" means signs never get lost.

Combine Like Terms 🧮

Simplify each expression. Enter only the coefficient of $x$ in the simplified result.

1) $6x + 3x$ → result is $\;?\,x$ 2) $9x - 4x$ → result is $\;?\,x$ 3) $5x + 2x - x$ → result is $\;?\,x$

The Distributive Property

To clear parentheses, multiply the outside factor by every term inside:

$a(b + c) = ab + ac$

Example — Expand $3(x + 4)$

$3(x + 4) = 3 \cdot x + 3 \cdot 4 = 3x + 12$

Example — Expand, then combine: $2(x + 5) + 3x$

$2(x + 5) + 3x = 2x + 10 + 3x = 5x + 10$

⚠️ Distribute to both terms. A frequent slip is $3(x + 4) = 3x + 4$ — the $3$ never reached the $4$. The factor must hit every term inside the parentheses.

Distribute & Simplify 🔽

Choose the correct simplified form of each expression.

🔤 Variables and Expressions

Part 5 of 5 — Real-World Modeling & Mastery Check

You can now (1) name the parts of an expression, (2) translate words into math, (3) evaluate by substituting, and (4) simplify by combining like terms and distributing. Let's put it all together and finish strong.

Building Expressions for Real Situations

A variable shines when a quantity can change. Pick a letter for the unknown, then describe the situation.

Example — A taxi charges $3 to start, plus $2 per mile.

Let $m$ = number of miles. The total cost (in dollars) is:

$3 + 2m$

A $5$-mile ride costs $3 + 2(5) = 3 + 10 = 13$, so $13.

Example — You have $20 and spend $4 each day.

Let $d$ = number of days. The money left (in dollars) is:

$20 - 4d$

After $3$ days you have $20 - 4(3) = 20 - 12 = 8$, so $8.

💡 The pattern: a fixed start-up amount becomes the constant, and the per-unit rate becomes the coefficient of the variable.

Apply It 🎯

Word-Problem Practice 🧮

A printer rents for a flat $10 fee plus $3 per hour. The cost is $10 + 3h$ dollars.

1) What is the cost for $h = 2$ hours? (enter just the number of dollars) 2) What is the cost for $h = 5$ hours? 3) What is the coefficient of $h$ in $10 + 3h$?

Mixed Review 🔽

One quick check pulled from each part of the lesson.

Quick Reference

⚠️ Top traps: "less than" flips the order; in $2x^2$ the square applies only to the $x$; $3x + 5x = 8x$ (the exponent never changes); and $3(x + 4) = 3x + 12$ — distribute to both terms.

Exit Quiz ✅

Answer all three to finish the lesson.