Variables and Algebraic Expressions

🔤 Variables and Algebraic Expressions

Part 1 of 5 — What Is a Variable?

Topics in This Part

🔑 Key Concept: A variable is just a letter that stands in for a number we don't know yet (or a number that's allowed to change). Algebra is arithmetic where some of the numbers are wearing name tags.

Variables and Constants

A variable is a symbol — usually a letter like $x$, $y$, $n$, or $t$ — that represents a number.

A constant is a fixed number that never changes, like $7$, $-3$, or $\dfrac{1}{2}$.

Why bother with letters? Because one expression can describe many situations at once. If a movie ticket costs $9, the cost of $n$ tickets is $9n$ — and that single rule works whether you buy $1$ ticket or $100$.

💡 The same variable can stand for different numbers in different problems, but within one problem each letter holds a single value at a time.

The Invisible Multiplication Sign

In algebra we almost never write the $\times$ symbol — it looks too much like the letter $x$. Instead, putting a number right next to a variable means multiply:

$3x \;=\; 3 \cdot x \;=\; 3 \text{ times } x$

The number stuck to the front, $3$, is called the coefficient.

⚠️ Watch out: $xy$ means $x \cdot y$, but $23$ does not mean $2 \cdot 3$ — two digits next to each other still make a normal two-digit number. The shortcut only applies when a letter is involved.

Concept Check 🎯

Powers and Order of Operations

An exponent is a shortcut for repeated multiplication:

$x^2 = x \cdot x \qquad x^3 = x \cdot x \cdot x$

We read $x^2$ as "$x$ squared" and $x^3$ as "$x$ cubed."

When an expression mixes operations, evaluate them in this order (PEMDAS):

💡 $2x^2$ means $2 \cdot (x^2)$ — the exponent attaches only to the $x$, not the $2$, because exponents come before multiplication. So if $x = 3$, then $2x^2 = 2 \cdot 9 = 18$, not $(2\cdot 3)^2 = 36$.

Translate to Numbers 🧮

Rewrite each as a plain number (no variables — just compute).

1) $4^2 = \,?$ 2) $2^3 = \,?$ 3) $5 + 3 \cdot 2 = \,?$ (remember order of operations)

Order of Operations Check 🔽

Choose the correct value of each numerical expression. Follow PEMDAS carefully.

🔤 Variables and Algebraic Expressions

Part 2 of 5 — Evaluating Expressions

🔑 The Idea: To evaluate an algebraic expression, you substitute a given number for each variable and then simplify using the order of operations. Replace the name tag with the actual number.

Substitution: Plug In and Compute

To evaluate an expression for a given value, swap the variable for its number — using parentheses to be safe — then follow PEMDAS.

Worked Example: evaluate $3x + 4$ when $x = 5$

$3x + 4 = 3(5) + 4 = 15 + 4 = 19$

Worked Example: evaluate $x^2 - 2x$ when $x = 4$

$x^2 - 2x = (4)^2 - 2(4) = 16 - 8 = 8$

⚠️ Use parentheses when you substitute, especially with negatives. For $x = -3$, writing $x^2 = (-3)^2 = 9$ keeps the sign right; writing $-3^2$ by accident gives $-9$, which is wrong here.

Substituting Negative Numbers

Worked Example: evaluate $5 - 2x$ when $x = -3$

$5 - 2x = 5 - 2(-3) = 5 - (-6) = 5 + 6 = 11$

Worked Example: evaluate $\dfrac{a + b}{2}$ when $a = 7$ and $b = 3$

$\frac{a + b}{2} = \frac{7 + 3}{2} = \frac{10}{2} = 5$

💡 A fraction bar acts like a set of parentheses: do all the arithmetic on top (and on the bottom) before you divide.

Concept Check 🎯

Evaluate It 🧮

Substitute and compute. Enter a single number for each.

1) $4x - 5$ when $x = 3 \;\Rightarrow\; ?$ 2) $\dfrac{12}{y} + 1$ when $y = 4 \;\Rightarrow\; ?$ 3) $a^2 - b$ when $a = 5,\; b = 9 \;\Rightarrow\; ?$

Match the Value 🔽

Each expression below is evaluated at $x = 2$. Pick the correct result.

🔤 Variables and Algebraic Expressions

Part 3 of 5 — Translating Words into Expressions

🔑 Why it matters: Real problems arrive as sentences, not equations. The skill of turning "seven more than a number" into $n + 7$ is the bridge from English to algebra.

The Keyword Dictionary

Certain words signal certain operations. Let $n$ be "the number."

⚠️ Order matters for subtraction. "$7$ less than a number" is $n - 7$ — the number comes first. Don't write $7 - n$. The phrase "less than" flips the order you read.

Translation Check 🎯

Building Multi-Step Expressions

Some phrases combine operations. Read carefully and group with parentheses when a quantity is acted on as a whole.

💡 The word "sum" or "difference" inside a longer phrase usually signals a set of parentheses: "$5$ times the sum of $n$ and $4$" must be $5(n+4)$, not $5n + 4$.

Phrase → Expression 🔽

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

From Words to Value 🧮

Translate, then evaluate at the given value.

1) "$5$ more than a number $n$," when $n = 8 \;\Rightarrow\; ?$ 2) "Three times a number $n$," when $n = 7 \;\Rightarrow\; ?$ 3) "$2$ less than a number $n$," when $n = 10 \;\Rightarrow\; ?$

🔤 Variables and Algebraic Expressions

Part 4 of 5 — Terms, Coefficients & Like Terms

🔑 Big Idea: Before you can simplify an expression, you have to break it into terms and spot which ones are "alike." Only like terms can be combined.

The Anatomy of an Expression

A term is a single number, variable, or product of them. Terms are separated by $+$ and $-$ signs.

Consider $\;4x + 7 - 2y$. It has three terms: $4x$, $7$, and $-2y$.

• The coefficient is the number multiplied by the variable.
• A constant term has no variable.
• The sign in front of a term belongs to that term, so the coefficient of $-2y$ is $-2$.

💡 If a variable has no visible number, its coefficient is $1$: the term $x$ means $1x$, and $-x$ means $-1x$.

Concept Check 🎯

What Makes Terms "Like"?

Like terms have the exact same variable part — same letters raised to the same powers. The coefficients can differ.

⚠️ Constants are like terms with each other. $7$ and $-3$ can combine to $4$. But a constant is never like a variable term.

Like or Not? 🔽

Decide whether each pair is a set of like terms.

Count and Identify 🧮

For the expression $\;9x - 4 + 3x + x\;$:

1) How many terms does it have? 2) Adding the coefficients of all the $x$-terms ($9$, $3$, and $1$), what total do you get?

🔤 Variables and Algebraic Expressions

Part 5 of 5 — Simplifying & Mastery Check

You can now name the parts of an expression and spot like terms. The payoff is simplifying: combining like terms and using the distributive property to write an expression in its cleanest form.

Combining Like Terms

To combine like terms, add or subtract their coefficients and keep the variable part unchanged.

Worked Example: simplify $7x + 2x$

$7x + 2x = (7 + 2)x = 9x$

Worked Example: simplify $5a + 3 - 2a + 8$

Group the like terms, then combine:

$(5a - 2a) + (3 + 8) = 3a + 11$

⚠️ You cannot combine $3a + 11$ any further — $3a$ is a variable term and $11$ is a constant. A simplified expression often still has more than one term, and that's fine.

The Distributive Property

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

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

Worked Example: expand $3(x + 4)$

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

Worked Example: simplify $2(n + 5) + 4n$

$2(n + 5) + 4n = 2n + 10 + 4n = (2n + 4n) + 10 = 6n + 10$

💡 Distribute first, then combine like terms. And watch the signs: $-2(x - 3) = -2x + 6$, because $-2 \cdot (-3) = +6$.

Simplify Check 🎯

Simplify It 🧮

Combine like terms and/or distribute. Enter the constant part of each simplified answer.

1) $5x + 7 + 2x - 3$ simplifies to $7x + ?$ (enter the constant) 2) $3(n + 6)$ expands to $3n + ?$ (enter the constant) 3) $2(a + 4) + a$ simplifies to $3a + ?$ (enter the constant)

Mixed Practice 🔽

Pull together everything from the lesson.

Exit Quiz ✅

Answer all three to finish the lesson.