Algebra Basics for Beginners: Variables, Expressions, and Solving Linear Equations (Step-by-Step)

Learn algebra fundamentals—what variables and expressions are, how to simplify and solve linear equations step-by-step, factoring basics, examples, practice problems, and common mistakes.

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Algebra: A Step-by-Step Guide

Algebra is the branch of mathematics that uses symbols (usually letters) to represent numbers and relationships between them. It lets us write general rules, solve problems, and work with unknowns.

1. Key concepts

  • Variable: a symbol (like x or y) that stands for an unknown number.
  • Constant: a fixed number (like 3, -5, or 1/2).
  • Expression: a combination of variables, constants, and operations (e.g., 3x + 2).
  • Equation: a statement that two expressions are equal (e.g., 3x + 2 = 11).
  • Coefficient: the number multiplied by a variable (in 4x, 4 is the coefficient).

2. Simplifying expressions

To simplify, combine like terms and use the distributive property.

  • Like terms: terms with the same variable part. Example: 5x and -2x can combine to 3x.
  • Distributive property: a(b + c) = ab + ac. Example: 3(x + 4) = 3x + 12.

Example

Simplify: 2(x + 3) + 4x

  1. Use distributive property: 2x + 6 + 4x
  2. Combine like terms: (2x + 4x) + 6 = 6x + 6

3. Solving linear equations (one variable)

Goal: isolate the variable (get x alone). Use inverse operations (undo addition with subtraction, undo multiplication with division).

One-step equations

Example: x + 5 = 12

  1. Subtract 5 from both sides: x + 5 - 5 = 12 - 5
  2. So x = 7

Two-step equations

Example: 3x - 4 = 11

  1. Undo -4 by adding 4: 3x - 4 + 4 = 11 + 4 → 3x = 15
  2. Undo multiplication by dividing: x = 15 / 3 = 5

Equations with parentheses and fractions

Example: 2(x + 3) = 14

  1. Distribute: 2x + 6 = 14
  2. Subtract 6: 2x = 8
  3. Divide by 2: x = 4

Example with fractions: (1/2)x + 3 = 7

  1. Subtract 3: (1/2)x = 4
  2. Multiply both sides by 2: x = 8

4. Checking your answer

Always substitute your solution back into the original equation to verify it works.

5. Factoring basics (useful for solving quadratic equations later)

Factoring rewrites an expression as a product. For example, x^2 + 5x + 6 factors to (x + 2)(x + 3) because 2 and 3 multiply to 6 and add to 5.

6. Worked examples

Example A

Solve 4x + 7 = 23

  1. Subtract 7: 4x = 16
  2. Divide by 4: x = 4
  3. Check: 4(4) + 7 = 16 + 7 = 23 ✔

Example B

Solve 5 - 2(x - 3) = 1

  1. Distribute: 5 - 2x + 6 = 1 → combine constants: 11 - 2x = 1
  2. Subtract 11: -2x = -10
  3. Divide by -2: x = 5
  4. Check: 5 - 2(5 - 3) = 5 - 2(2) = 5 - 4 = 1 ✔

7. Practice problems

  1. Solve: x + 9 = 15
  2. Solve: 2x - 7 = 9
  3. Simplify: 3(x + 2) + 4x
  4. Solve: (1/3)x + 5 = 8
  5. Factor: x^2 + 7x + 10

8. Solutions (step-by-step)

  1. x + 9 = 15 → subtract 9 → x = 6
  2. 2x - 7 = 9 → add 7 → 2x = 16 → divide 2 → x = 8
  3. 3(x + 2) + 4x → 3x + 6 + 4x = 7x + 6
  4. (1/3)x + 5 = 8 → subtract 5 → (1/3)x = 3 → multiply by 3 → x = 9
  5. x^2 + 7x + 10 → factors of 10 that add to 7 are 5 and 2 → (x + 5)(x + 2)

9. Tips and common mistakes

  • Do the same operation to both sides of an equation.
  • Combine like terms before solving when possible.
  • Watch signs: subtracting a negative is adding, and distributing negative signs matters.
  • Check your solution by substituting back into the original equation.

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