Expressions & Equations

1. Introduction

So far in math you have mostly worked with numbers you can see — like $3 + 5$ or $12 \div 4$. Algebra adds one powerful new idea: a letter can stand in for a number we do not know yet. That letter is called a variable, and once we allow variables, we can describe patterns, write rules, and solve real puzzles like "I'm thinking of a number; if I double it and add $3$ I get $11$ — what is it?"

This article covers two closely related skills. First, working with expressions — combinations of numbers, variables, and operations like $3x + 5$. We will learn to evaluate them (plug in a number) and simplify them (write them more neatly). Second, solving equations — statements that two expressions are equal, like $2x + 3 = 11$. Solving an equation means finding the value of the variable that makes the statement true.

The single most important idea you will take away is the balance principle: an equation is like a balanced scale, and whatever you do to one side you must do to the other to keep it balanced. Master that one idea and equations stop being scary forever. Along the way you will also meet the distributive property, like terms, and the order of operations — tools that make every step predictable.

Algebra is the language of patterns. The expression $3x + 5$ is not just a random string of symbols; it is a rule that says "take any number $x$, triple it, and add $5$." When you evaluate at $x = 4$, you get $17$. When you solve $3x + 5 = 17$, you are asking "which input makes the rule output $17$?" That input is $x = 4$. Expressions and equations are two sides of the same coin.

In grade $7$, you are expected to fluently move between numeric computation and symbolic reasoning. That means you can simplify $4x + 2x$ to $6x$ without drawing boxes every time, but you still understand why it works. It also means you can set up and solve equations that model everyday situations — distances, ages, costs, and unknown counts. The skills in this chapter are the backbone of every algebra course that follows.

2. Core Concepts

Variables, terms, and expressions

A variable is a letter (often $x$, $y$, or $n$) that represents a number. A constant is a fixed number like $7$. A coefficient is the number multiplied by a variable; in $4x$, the coefficient is $4$. A term is a single number, a single variable, or a number times variables — the pieces separated by $+$ and $-$ signs. For example, in

$3x + 5y - 7,$

the terms are $3x$, $5y$, and $-7$. An expression is the whole combination of terms. Notice an expression has no equals sign — it is a phrase, not a sentence.

Evaluating an expression

To evaluate an expression means to replace each variable with a given number and then compute the result. If $x = 4$, then $3x + 5$ becomes $3(4) + 5 = 12 + 5 = 17$. The expression $3x + 5$ is a rule; evaluating it at $x = 4$ gives the single number $17$.

This is where the order of operations matters. We compute in this order: Parentheses, Exponents, Multiplication and Division (left to right), then Addition and Subtraction (left to right) — often remembered as PEMDAS. Multiplication happens before addition, which is why $3(4) + 5$ is $17$ and not $3 \times 9 = 27$.

Like terms and why we can combine them

Like terms are terms that have exactly the same variable part. So $4x$ and $2x$ are like terms, and $5y$ and $-y$ are like terms, but $4x$ and $2x^2$ are not (different powers), and $4x$ and $4y$ are not (different variables).

Why can we combine like terms? Think of $x$ as a box. If you have $4$ boxes plus $2$ more boxes, you have $6$ boxes — that is exactly $4x + 2x = 6x$. But $4$ boxes plus $2$ crates cannot merge into a single count, just as $4x + 2y$ cannot be simplified. We only add the coefficients of like terms; the variable part stays the same.

The distributive property

The distributive property says $a(b + c) = ab + ac$. It tells us how multiplication interacts with a sum inside parentheses: the outside factor is multiplied by every term inside. For instance,

$3(x + 2) = 3 \cdot x + 3 \cdot 2 = 3x + 6.$

Picture $3$ identical bags, each holding $x$ apples and $2$ oranges. Altogether you have $3x$ apples and $6$ oranges — that is exactly $3x + 6$. The distributive property also works with subtraction: $3(x - 2) = 3x - 6$, and with a negative factor: $-2(x + 4) = -2x - 8$.

What an equation is — and what "solving" means

An equation sets two expressions equal with an equals sign, like $2x + 3 = 11$. The equals sign is a promise that the left side and the right side have the same value. Solving an equation means finding every value of the variable that keeps that promise true. For $2x + 3 = 11$, the solution is $x = 4$, because $2(4) + 3 = 11$ really is true. Plugging $x = 4$ back in to confirm is called checking, and it is the best way to catch mistakes.

The balance principle and inverse operations

An equation behaves like a balanced scale: the two sides weigh the same. If you add the same weight to both pans, or remove the same weight from both, the scale stays balanced. In algebra this is the balance principle: whatever operation you do to one side, you must do to the other.

To get the variable alone, we "undo" operations using their inverse (opposite): addition undoes subtraction, multiplication undoes division. To peel away a $+3$, we subtract $3$; to peel away a $\times 2$, we divide by $2$. We always undo in the reverse order of operations.

One-step versus two-step equations

A one-step equation requires a single inverse operation to isolate the variable, like $x + 5 = 12$ (subtract $5$) or $3x = 21$ (divide by $3$). A two-step equation needs two undo operations, like $2x + 3 = 11$: first undo the addition, then undo the multiplication. The key is always working in reverse PEMDAS order — peel off addition and subtraction before multiplication and division.

Translating words into algebra

Word problems become manageable when you translate phrase by phrase. "A number" becomes a variable like $n$. "Multiply by $3$" becomes $3n$. "Subtract $7$" becomes $-7$. "Is $20$" becomes $= 20$. The sentence "I multiply a number by $3$ and subtract $7$ to get $20$" becomes $3n - 7 = 20$. Building the equation is often harder than solving it — take it one phrase at a time.

Checking your solution

After solving, substitute your answer back into the original equation (before you simplified or distributed). If both sides are equal, your solution is correct. If they are not, retrace your steps. Checking takes ten seconds and catches most arithmetic slips.

Coefficients of $1$ and $-1$

When a variable appears alone, its coefficient is $1$: the expression $x + 5$ really means $1x + 5$. When it appears with a minus sign, the coefficient is $-1$: $-x + 5$ means $-1x + 5$. This matters when combining like terms: $x - 3x = 1x - 3x = -2x$. Writing the invisible $1$ in your head prevents sign mistakes.

Exponents in expressions

An exponent tells you how many times to multiply a base by itself. In $x^2$, the base is $x$ and the exponent is $2$, so $x^2 = x \cdot x$. When evaluating $2x^2$ at $x = 3$, the exponent applies only to the $3$: $2(3)^2 = 2(9) = 18$, not $(2 \cdot 3)^2 = 36$. Exponents come before multiplication in PEMDAS.

Writing solutions clearly

A solution to an equation is a value, not just a number floating on the page. Write $x = 4$, not just $4$. In word problems, finish with a sentence: "The number is $9$." Clear communication is part of good mathematics.