Numbers and Algebra: Algebra

Key Concepts

• Algebra is a branch of Mathematics where letters are used to represent numbers.
• A letter in algebra can stand for:
  • an unknown value we need to find
  • a value that can change
  • a general rule or pattern

Using letters to represent unknown values

• We use letters such as x, y, n, a, b to stand for numbers.

• Example:

  • If Ali has some marbles and we do not know how many, we can say he has m marbles.

• A letter helps us write:

  • number patterns
  • rules
  • word problems more easily

• Example:

  • “A number plus 5” can be written as n + 5
  • “3 times a number” can be written as 3n

Terms, coefficients and constants

• An algebraic expression is a mathematical phrase with numbers, letters and operations.
  • Example: 3x + 4
• A term is a part of an expression separated by + or −.
  • In 3x + 4, the terms are 3x and 4
• A coefficient is the number multiplying a letter.
  • In 3x, the coefficient is 3
• A constant is a number on its own, with no letter.
  • In 3x + 4, the constant is 4

Simplifying algebraic expressions

• To simplify means to write an expression in its shortest and clearest form.
• We can simplify by:
  • combining like terms
  • removing unnecessary signs
  • writing repeated multiplication properly

Like terms

• Like terms are terms with the same letter part.
  • Example: 3x and 5x are like terms
  • Example: 2a and 7a are like terms
• We can add or subtract like terms.
  • 3x + 5x = 8x
  • 9a - 4a = 5a
• We cannot combine unlike terms.
  • 3x + 2y cannot become 5xy
  • 4a + 3 cannot become 7a

Writing multiplication in algebra

• In algebra, we usually do not write the multiplication sign between a number and a letter.
  • 3 × x = 3x
  • a × b = ab
• Be careful:
  • 2a means 2 × a
  • it does not mean 20 + a

Evaluating algebraic expressions

• To evaluate an expression means to find its value when the letters are given.

• Steps:

  1. substitute the given value into the expression
  2. use brackets if needed
  3. carry out the operations carefully

• Example:

  • Evaluate 3x + 2 when x = 4
  • Substitute: 3(4) + 2
  • Calculate: 12 + 2 = 14

Solving simple linear equations

• An equation is a mathematical statement showing that two sides are equal.

  • Example: x + 5 = 12

• A simple linear equation has:

  • one unknown
  • the unknown is not squared or cubed
  • the highest power of the unknown is 1

• To solve an equation means to find the value of the unknown that makes the equation true.

Balance idea in equations

• Think of an equation like a balance scale.

• Both sides must stay equal.

• Whatever you do to one side, you must do the same to the other side.

• Example:

  • If x + 3 = 10
  • subtract 3 from both sides
  • x = 7

Inverse operations

• Inverse operations undo each other.

  • addition and subtraction
  • multiplication and division

• We use inverse operations to solve equations.

• Example:

  • x + 8 = 15
  • subtract 8 from both sides
  • x = 7

• Example:

  • 4x = 20
  • divide both sides by 4
  • x = 5

Algebra word problems

• In word problems, algebra helps us represent unknown amounts clearly.

• Usual steps:

  1. choose a letter for the unknown
  2. write the information as an equation or expression
  3. solve carefully
  4. check if the answer makes sense in the context

• Keywords to notice:

  • sum means add
  • difference means subtract
  • product means multiply
  • quotient means divide
  • more than means add
  • less than means subtract
  • twice means multiply by 2
  • three times means multiply by 3

Translating words into algebra

• “A number increased by 6” → n + 6
• “5 less than a number” → n - 5
• “Twice a number” → 2n
• “A number divided by 4” → n ÷ 4 or n/4
• “The total of x and 7” → x + 7

Two-Step Linear Equations

• One-step recap: equations such as 4x = 20 need just one step — divide both sides by 4.

• Two-step equations need two operations to isolate the unknown.

  • Example: 2x + 3 = 11
    • Step 1: Subtract 3 from both sides first (undo the addition).
    • Step 2: Divide both sides by 2 (undo the multiplication).
  • Always undo addition or subtraction first, then undo multiplication or division.

• Equations with brackets: expand the brackets OR divide first, then solve.

  • Example: 3(x + 2) = 18
    • Method 1 (expand): 3x + 6 = 18 → subtract 6 → 3x = 12 → divide by 3 → x = 4
    • Method 2 (divide first): divide both sides by 3 → x + 2 = 6 → subtract 2 → x = 4

Worked Example: Solve 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

Step 3: Check

• Substitute x = 4: 2(4) + 3 = 8 + 3 = 11 ✔

Answer: x = 4

Worked Example: Solve 3(x + 2) = 18

Step 1: Divide both sides by 3

• 3(x + 2) ÷ 3 = 18 ÷ 3
• x + 2 = 6

Step 2: Subtract 2 from both sides

• x + 2 − 2 = 6 − 2
• x = 4

Step 3: Check

• Substitute x = 4: 3(4 + 2) = 3(6) = 18 ✔

Answer: x = 4

Checking answers

• After solving, always substitute your answer back into the equation.
• Example:
  • If x = 5 for 2x + 1 = 11
  • Check: 2(5) + 1 = 10 + 1 = 11
  • Correct

Important Definitions

• Algebra: a branch of Mathematics that uses letters and symbols to represent numbers and relationships.
• Variable: a letter that represents an unknown or changing value.
• Unknown: a value that is not yet known and must be found.
• Algebraic expression: a mathematical phrase made up of numbers, letters and operations, without an equals sign.
• Term: a part of an algebraic expression separated by addition or subtraction signs.
• Coefficient: the number multiplying a variable.
• Constant: a number without any variable.
• Like terms: terms with the same variable part.
• Simplify: to write an expression in a shorter or easier form without changing its value.
• Substitute: to replace a variable with a given number.
• Evaluate: to find the value of an expression after substituting the given number.
• Equation: a mathematical statement showing that two expressions are equal.
• Linear equation: an equation in which the variable has a power of 1.
• Inverse operations: operations that undo each other, such as addition and subtraction, or multiplication and division.
• Solution: the value of the variable that makes an equation true.

Worked Examples

Example 1: Simplifying an algebraic expression

Simplify 3a + 5a - 2

Step 1: Identify like terms

• 3a and 5a are like terms because both have a
• 2 is a constant

Step 2: Combine the like terms

• 3a + 5a = 8a

Step 3: Write the simplified expression

• 8a - 2

Answer: 8a - 2

Example 2: Evaluating an algebraic expression

Evaluate 2x + 3 when x = 6

Step 1: Substitute the value of x

• 2x + 3 = 2(6) + 3

Step 2: Multiply first

• 2(6) = 12

Step 3: Add

• 12 + 3 = 15

Answer: 15

Example 3: Solving a simple linear equation

Solve 3x = 21

Step 1: Identify the operation

• x is being multiplied by 3

Step 2: Use the inverse operation

• Divide both sides by 3

Step 3: Simplify

• x = 7

Step 4: Check

• Substitute x = 7
• 3(7) = 21
• 21 = 21, so the answer is correct

Answer: x = 7

Example 4: Algebra word problem

Sam has 4 more stickers than Ben. Sam has 15 stickers. How many stickers does Ben have?

Step 1: Let the unknown be a letter

• Let b be the number of stickers Ben has.

Step 2: Write Sam’s number using b

• Sam has b + 4 stickers

Step 3: Form the equation

• b + 4 = 15

Step 4: Solve the equation

• Subtract 4 from both sides:
• b = 15 - 4
• b = 11

Step 5: Write the answer clearly

• Ben has 11 stickers.

Step 6: Check

• If Ben has 11 stickers, Sam has 11 + 4 = 15
• This matches the question

Answer: Ben has 11 stickers.

Common Mistakes to Avoid

• Combining unlike terms wrongly
  • Wrong: 3x + 2 = 5x
  • Correct: 3x + 2 cannot be simplified further
• Forgetting to substitute properly
  • Wrong: if x = 4, writing 3x = 34
  • Correct: 3x = 3(4) = 12
• Not using brackets when substituting
  • Example: for 2x + 3, write 2(4) + 3, not 24 + 3
• Doing different operations on the two sides of an equation
  • Both sides must stay balanced
• Forgetting inverse operations
  • To undo +5, subtract 5
  • To undo ×3, divide by 3
• Mixing up 5 less than a number and a number less 5
  • Both mean n - 5
• Misreading “less than”
  • “5 less than 12” means 12 - 5, not 5 - 12
• Forgetting to check the answer in the original equation
• Leaving the answer without units or context in word problems
  • Example: write 11 stickers, not just 11
• Writing multiplication signs in confusing ways
  • Avoid writing x × x if it may be mistaken for the variable x
  • Use clear notation like 3a, 2b

Exam Tips

• Always let the unknown be a letter first in word problems.
  • Example: “Let x be the number of apples.”
• Translate the words carefully before calculating.
• Look for operation keywords:
  • total, sum → add
  • difference → subtract
  • times, twice → multiply
  • shared equally, quotient → divide
• When simplifying:
  • combine only like terms
• When solving equations:
  • show the operation clearly
  • keep both sides equal
• When evaluating:
  • substitute the value correctly
  • use brackets if needed
• Always check:
  • Does the answer fit the question?
  • Is the value reasonable?
• In word problems:
  • write a final statement
  • Example: “Therefore, Ben has 11 stickers.”
• If marks are given for working, do not skip steps.
• Neat and correct algebra helps avoid careless mistakes.

Useful mark-earning phrases:

• “Let x be the number of …”
• “Therefore,…”
• “Substitute x = …”
• “Check: …”
• “The equation is …”

Quick Summary

• Algebra uses letters to represent unknown or changing numbers.
• A variable is the letter used to stand for a number.
• An algebraic expression has numbers, letters and operations, but no equals sign.
• An equation has an equals sign and can be solved.
• Terms are separated by + or − signs.
• A coefficient is the number multiplying the variable.
• A constant is a number without a variable.
• Combine only like terms when simplifying.
• To evaluate, substitute the given value into the expression and calculate.
• To solve equations, use inverse operations and keep both sides balanced.
• In word problems, let a letter represent the unknown, form an equation, solve, and check.
• Always give the final answer clearly, with units or labels when needed.