Algebra Basics | StudyHelper Hub

What Is Algebra?

Arithmetic deals with specific numbers: 3 + 4 = 7. Algebra takes the next step by using letters to represent numbers we do not yet know. Instead of asking "what is 3 + 4?", algebra asks "if something plus 4 equals 7, what is that something?"

That unknown "something" is what algebra calls a variable. Variables let us write general rules that work for any number, not just one specific case. This makes algebra incredibly powerful for solving real problems.

Variables and Expressions

A variable is a letter that stands in for an unknown number. The most common variables are x, y, and n, but any letter works. When you see "x" in a math problem, your job is usually to figure out what number x represents.

A mathematical expression is a combination of numbers, variables, and operations (like + or ×). Expressions do not have an equals sign. They just represent a value.

Examples of Expressions

x + 5 — a number plus five
3n — three times some number (note: 3n means 3 × n; the multiplication sign is usually left out)
2y - 7 — twice a number, minus seven
x / 4 — a number divided by four

An equation is two expressions joined by an equals sign. It makes a statement: the left side equals the right side. Your job when solving an equation is to find the value of the variable that makes that statement true.

Key Vocabulary

Variable — a letter representing an unknown number
Coefficient — the number multiplied by a variable (in 3x, the coefficient is 3)
Constant — a plain number with no variable attached
Term — a single piece of an expression (3x, 7, and y are each terms)
Like terms — terms with the same variable, which can be combined (3x + 5x = 8x)

Order of Operations (PEMDAS)

When a math expression has multiple operations, you must follow a specific order. Without this rule, different people solving the same problem could get different answers. The order of operations tells everyone to solve problems the same way.

The acronym PEMDAS helps you remember the order:

  P  -- Parentheses         (solve what is inside brackets first)
  E  -- Exponents           (powers and square roots)
  M  -- Multiplication  \
                          (left to right, whichever comes first)
  D  -- Division        /
  A  -- Addition        \
                          (left to right, whichever comes first)
  S  -- Subtraction     /

Memory Trick Please Excuse My Dear Aunt Sally = Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. Note that M and D have equal priority (same for A and S) — just work left to right for those pairs.

PEMDAS Step-by-Step Example

Solve: 3 + 2 × (8 - 5)² ÷ 3

  Step 1 — Parentheses:   8 - 5 = 3
  Expression becomes:     3 + 2 × 3² ÷ 3

  Step 2 — Exponents:     3² = 9
  Expression becomes:     3 + 2 × 9 ÷ 3

  Step 3 — Multiply/Divide (left to right):
             2 × 9 = 18   then   18 ÷ 3 = 6
  Expression becomes:     3 + 6

  Step 4 — Addition:      3 + 6 = 9

  Answer: 9

Solving One-Step Equations

A one-step equation needs only a single operation to isolate the variable. The key principle is this: whatever you do to one side of the equation, you must do to the other side. This keeps the equation balanced.

Think of an equation as a perfectly balanced scale. If you add weight to one side, you must add the same weight to the other side or it tips over. The same logic applies to equations.

One-Step Examples

  Example 1: x + 9 = 15
  Goal: get x alone. Subtract 9 from both sides.
  x + 9 - 9 = 15 - 9
  x = 6
  Check: 6 + 9 = 15 ✓

  Example 2: y - 4 = 11
  Goal: get y alone. Add 4 to both sides.
  y - 4 + 4 = 11 + 4
  y = 15
  Check: 15 - 4 = 11 ✓

  Example 3: 5n = 35
  Goal: get n alone. Divide both sides by 5.
  5n ÷ 5 = 35 ÷ 5
  n = 7
  Check: 5 × 7 = 35 ✓

  Example 4: x / 3 = 8
  Goal: get x alone. Multiply both sides by 3.
  (x / 3) × 3 = 8 × 3
  x = 24
  Check: 24 / 3 = 8 ✓

Solving Two-Step Equations

Two-step equations require two operations to isolate the variable. The strategy is to work in reverse order of PEMDAS: deal with addition or subtraction first, then multiplication or division.

Two-Step Examples

  Example 1: 2x + 3 = 11
  Step 1 — Subtract 3 from both sides:
    2x + 3 - 3 = 11 - 3
    2x = 8
  Step 2 — Divide both sides by 2:
    2x ÷ 2 = 8 ÷ 2
    x = 4
  Check: 2(4) + 3 = 8 + 3 = 11 ✓

  Example 2: 3y - 7 = 14
  Step 1 — Add 7 to both sides:
    3y - 7 + 7 = 14 + 7
    3y = 21
  Step 2 — Divide both sides by 3:
    3y ÷ 3 = 21 ÷ 3
    y = 7
  Check: 3(7) - 7 = 21 - 7 = 14 ✓

  Example 3: n/4 + 5 = 9
  Step 1 — Subtract 5 from both sides:
    n/4 = 4
  Step 2 — Multiply both sides by 4:
    n = 16
  Check: 16/4 + 5 = 4 + 5 = 9 ✓

Always Check Your Answer After solving, plug your answer back into the original equation. If both sides are equal, you solved it correctly. This takes 10 seconds and catches most errors.

Translating Word Problems into Equations

Word problems describe a situation in plain English. Your job is to translate that description into an algebraic equation, then solve it. The trick is recognizing the math words hidden in the sentences.

Math Words to Look For

is, equals, is the same as → = (the equals sign)
sum, more than, added to, increased by, plus → +
difference, less than, decreased by, minus, fewer → −
product, times, of, multiplied by, twice → ×
quotient, divided by, per, ratio of → ÷
a number, some number, an unknown → x (or any variable)

Word Problem Walk-Through

Problem: "A number is multiplied by 6, then 4 is subtracted. The result is 20. What is the number?"

  Step 1 — Name the unknown:
    Let x = the unknown number

  Step 2 — Translate piece by piece:
    "A number is multiplied by 6" → 6x
    "then 4 is subtracted"        → 6x - 4
    "The result is 20"            → 6x - 4 = 20

  Step 3 — Solve:
    6x - 4 + 4 = 20 + 4
    6x = 24
    x = 4

  Step 4 — Check:
    6(4) - 4 = 24 - 4 = 20 ✓

  Answer: The number is 4.

Another Word Problem

Problem: "Emma has some money. She earns $15 more babysitting and now has $47. How much did she start with?"

  Let x = starting amount
  "Earns $15 more" → x + 15
  "Now has $47"    → x + 15 = 47

  Solve:
    x + 15 - 15 = 47 - 15
    x = 32

  Emma started with $32.
  Check: 32 + 15 = 47 ✓

Introduction to Inequalities

An inequality is like an equation, but instead of saying two things are equal, it says one is larger or smaller than the other. Inequalities use these symbols:

  >   greater than          (x > 5 means x is more than 5)
  <   less than             (x < 3 means x is less than 3)
  ≥   greater than or equal (x ≥ 7 means x is 7 or more)
  ≤   less than or equal    (x ≤ 10 means x is 10 or less)
  ≠   not equal to          (x ≠ 0 means x is anything but 0)

You solve inequalities almost exactly the same way you solve equations, with one important exception: if you multiply or divide both sides by a negative number, you must flip the inequality sign.

Solving an Inequality

  Solve: 2x + 1 < 9

  Step 1 — Subtract 1 from both sides:
    2x < 8

  Step 2 — Divide both sides by 2:
    x < 4

  This means x can be any number less than 4.
  (e.g., 3, 2.5, 0, -100 all work)

  --- Negative number example ---
  Solve: -3x > 12

  Divide both sides by -3 (FLIP the sign!):
    x < -4

Working with Fractions in Algebra

Fractions appear regularly in algebra. If you need a refresher on how fractions work, visit our Fractions guide. In algebra, fractions with variables follow the same rules as regular fractions — you just treat the variable like a number.

Useful Connection Percentages are really a form of algebra. When you calculate "what is 30% of 80?", you are solving x = 0.30 × 80. See our Percentages guide for more examples where algebra and percentages connect.

Tips for Algebra Success

Show every step — do not try to do multiple steps in your head; write each one out
Check your answer — plug it back into the original equation every time
Keep the equation balanced — any operation you apply to one side must be applied to the other
Combine like terms first — if you see 3x + 2x, combine them to 5x before doing anything else
Translate slowly — with word problems, go phrase by phrase, not the whole sentence at once
Practice regularly — algebra is a skill; it gets easier the more problems you work through

Frequently Asked Questions

Why do we use letters instead of just numbers?

Letters let us write rules that apply to any number, not just one specific value. For example, the rule "the area of a rectangle is length times width" can be written as A = l × w. That formula works for every rectangle ever — you just plug in different values of l and w. Without variables, you would need a separate calculation for every single rectangle.

What does it mean to "simplify" an expression?

Simplifying means rewriting an expression in its shortest, cleanest form without changing its value. This usually involves combining like terms (3x + 5x = 8x) and applying the order of operations. A simplified expression is easier to work with and less likely to cause errors in later steps.

Can a variable equal a fraction or a negative number?

Absolutely. Variables can represent any real number — positive, negative, zero, whole numbers, fractions, or decimals. There is no restriction unless the problem specifically states one (like "x must be a whole number"). Always let the math tell you what the variable is rather than assuming it has to be a "nice" number.

What is the difference between an expression and an equation?

An expression has no equals sign — it is just a mathematical phrase, like 3x + 2. You can simplify an expression but you cannot "solve" it because there is no specific value being claimed. An equation has an equals sign (3x + 2 = 14) and makes a claim. You solve an equation by finding the value of x that makes the claim true.

Why does dividing by a negative number flip the inequality sign?

Think about this: 2 < 6 is true. Now multiply both sides by -1: -2 and -6. On the number line, -6 is to the left of -2, meaning -6 is actually smaller than -2. So the relationship flips: -2 > -6. The same flip happens whenever you multiply or divide an inequality by any negative number. Always watch for this — it is one of the most common algebra mistakes.

Quick Quiz

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