Introduction to Algebra

A variable is a letter (like $x$, $n$, or $y$) that stands for a number we do not know yet, or a number that can change. Instead of writing “some number,” we just write a letter. The same letter always stands for the same value within one problem.

An expression is a combination of numbers, variables, and operations with no equals sign, such as $3x + 7$. An equation says two expressions are equal and does have an equals sign, such as $3x + 7 = 19$. You simplify or evaluate an expression; you solve an equation.

Consider $\ 5x + 2y - 8$.

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• A term is a piece separated by $+$ or $-$ signs: the terms are $5x$, $2y$, and $-8$.
• A coefficient is the number multiplied by a variable: $5$ is the coefficient of $x$, and $2$ is the coefficient of $y$.
• A constant is a term that is just a number, with no variable: here the constant is $-8$.

Certain words signal each operation:

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• Addition: sum, plus, more than, increased by, total
• Subtraction: difference, minus, less than, decreased by, fewer
• Multiplication: product, times, of, twice, double, triple
• Division: quotient, divided by, per, split equally

To evaluate an expression, replace each variable with its given value and follow the order of operations (PEMDAS). Wrap negative values in parentheses so you do not lose a sign.

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• Commutative (order can swap): $a + b = b + a$ and $ab = ba$.
• Associative (regroup with parentheses): $(a+b)+c = a+(b+c)$ and $(ab)c = a(bc)$.
• Distributive (multiply across a sum): $a(b+c) = ab + ac$.
• Identity (leaves a number unchanged): $a + 0 = a$ and $a \times 1 = a$.

To expand, multiply the number outside the parentheses by each term inside, keeping track of signs:

Like terms have the exact same variable part (same letter, same power). You combine them by adding or subtracting their coefficients. Constants are like terms with each other. You may not combine terms with different variables.

To fully simplify an expression: first use the distributive property to remove any parentheses, then combine all like terms. Work carefully with signs at each step.

A function is a rule that takes an input and produces exactly one output. Instead of naming the rule “the expression $3x + 5$,” we give it a name and write function notation: $f(x) = 3x + 5$. Read this as “$f$ of $x$.” The letter inside the parentheses is the input; the whole symbol $f(x)$ stands for the output. Note that $f(x)$ does not mean $f$ times $x$---it is the single value the rule returns.

A literal equation has several letters, and we solve for one of them in terms of the others. The strategy is the same as solving a numerical equation: use inverse operations to isolate the wanted letter, treating every other letter as if it were a known number.

An inequality uses $<$, $>$, $\le$, or $\ge$ instead of an equals sign, and describes a whole range of values. You solve it almost exactly like an equation---add, subtract, multiply, and divide on both sides---with one crucial exception: whenever you multiply or divide both sides by a negative number, you must reverse the inequality symbol.

The distributive property $a(b + c) = ab + ac$ can be run backward. Factoring means pulling out a common factor shared by every term: $ab + ac = a(b + c)$. To factor, find the greatest common factor (GCF) of all the terms, write it outside the parentheses, and put what remains inside.

Real problems become algebra in three moves: (1) let a letter stand for the unknown quantity; (2) translate each sentence into an expression or equation; (3) solve, then check the answer against the words. When two quantities are related, define both in terms of the same variable.

Many number patterns grow by the same amount each step. If a sequence starts at a first value and adds a fixed amount $d$ every term, the $n$th term has a closed-form rule: (first term) $+\ (n - 1)\cdot d$. A closed form lets you jump straight to any term without listing them all.