Algebra (Intro) — Class 6 IMO

Variables, Constants & Expressions

What is an Algebraic Expression?

An algebraic expression is a mathematical phrase that contains numbers, variables (letters), and operations (+, -, ×, ÷). It does NOT have an equals sign.

Variable: A letter that represents an unknown number (like x, y, n, a)

Key Words for Operations:

Operation	Key Words	Example	Expression
Addition (+)	plus, more than, sum, increased by, total	5 more than a number	x + 5
Subtraction (-)	minus, less than, difference, decreased by, fewer	3 less than a number	x - 3
Multiplication (×)	times, product, of, twice, triple	twice a number	2x
Division (÷)	divided by, quotient, half, split	half of a number	x/2 or x÷2

Important Order Notes:

"5 more than a number" → x + 5
"5 less than a number" → x - 5 (NOT 5 - x!)
"Twice a number" → 2x
"A number divided by 3" → x/3

TRANSLATING WORDS TO EXPRESSIONS - FLOWCHART:

    Word Phrase
         │
         ▼
    Identify the variable
    (what is unknown?)
         │
         ▼
    Identify the operation
    (look for key words)
         │
         ▼
    Write in correct order
    (watch for "less than"!)
         │
         ▼
    Algebraic Expression


COMMON PHRASES AND THEIR MEANINGS:

    "more than" → ADD (x + 5)
    "less than" → SUBTRACT but REVERSE! (x - 5, not 5 - x)
    "times" → MULTIPLY (3x)
    "divided by" → DIVIDE (x/4)
    "the sum of" → ADD (x + y)
    "the difference of" → SUBTRACT (x - y)
    "the product of" → MULTIPLY (x × y)
    "the quotient of" → DIVIDE (x ÷ y)


PRACTICE - FILL IN THE TABLE:

    Words                    Expression
    ──────────────────────────────────────────
    7 more than n            n + 7
    7 less than n            n - 7
    n more than 7            7 + n
    n less than 7            7 - n  (careful!)
    twice n                  2n
    n divided by 5           n/5
    4 times the sum of n and 3   4(n + 3)

Example 1: In 3x + 5, name the variable and the constant.

Variable = x; constant = 5.

Example 2: Write 'five more than x' as an expression.

x + 5.

Example 3:

Write an expression for "5 more than twice a number"

"a number" = let it be x
"twice a number" = 2x
"5 more than" = add 5
Answer: 2x + 5

Example 4:

Write an expression for "7 less than a number"

"a number" = let it be n
"7 less than" means subtract 7 FROM the number
Answer: n - 7 (NOT 7 - n!)

Example 5:

Write an expression for "the sum of 3 times a number and 8"

"a number" = let it be y
"3 times a number" = 3y
"the sum of ... and 8" = add 8
Answer: 3y + 8

Example 6:

Write an expression for "four less than the product of 6 and a number"

"a number" = let it be a
"product of 6 and a number" = 6a
"four less than" = subtract 4
Answer: 6a - 4

Quick recap

Variable = a letter for an unknown; constant = a fixed number.
Turn words into expressions (twice n = 2n).
Variable = unknown number (use x, n, a, etc.)
"More than" and "less than" are DIFFERENT orders
"Less than" flips the order: n - 5, not 5 - n
Multiplication: 2x means 2 × x
Parentheses group operations: 3(x + 4)

✓ Quick check

In 2y + 7, what is the constant?

7 is the fixed number, so it is the constant.

Write 'twice a number n' as an expression.

Twice n means 2 × n = 2n.

Evaluating Expressions

What Does "Evaluate" Mean?

To evaluate an expression means to substitute (replace) the variable with a number and then calculate the result.

Steps to Evaluate:

Write the expression
Replace each variable with its given value (use parentheses!)
Follow order of operations (PEMDAS)
Simplify to get a single number

Example: Evaluate 3x + 4 when x = 6

Write: 3x + 4
Substitute: 3(6) + 4
Calculate: 18 + 4 = 22

EVALUATING EXPRESSIONS - STEP BY STEP:

    Expression: 2n + 5, where n = 7
    
    Step 1: Write expression    2n + 5
    Step 2: Substitute n=7      2(7) + 5
    Step 3: Multiply first      14 + 5
    Step 4: Add                  19
    
    Answer: 19


EVALUATING WITH MULTIPLE VARIABLES:

    Expression: 3a + 2b, where a = 4, b = 5
    
    Step 1: 3a + 2b
    Step 2: 3(4) + 2(5)
    Step 3: 12 + 10
    Step 4: 22
    
    Answer: 22


EVALUATING WITH PARENTHESES:

    Expression: 2(x + 3), where x = 4
    
    Step 1: 2(x + 3)
    Step 2: 2(4 + 3)
    Step 3: 2(7) = 14
    
    Answer: 14


EVALUATING WITH EXPONENTS:

    Expression: x² + 3x - 2, where x = 5
    
    Step 1: x² + 3x - 2
    Step 2: 5² + 3(5) - 2
    Step 3: 25 + 15 - 2
    Step 4: 40 - 2 = 38
    
    Answer: 38

Example 1: If x = 4, find the value of 3x.

3 × 4 = 12.

Example 2: If a = 5, find the value of a + 7.

5 + 7 = 12.

Example 3:

Evaluate 4x - 7 when x = 9

Substitute: 4(9) - 7
Multiply: 36 - 7
Answer: 29

Example 4:

Evaluate 2a + 3b - c when a = 5, b = 4, c = 2

Substitute: 2(5) + 3(4) - 2
Multiply: 10 + 12 - 2
Add/Subtract: 22 - 2 = 20
Answer: 20

Example 5:

Evaluate 5(n - 3) + 2 when n = 10

Substitute: 5(10 - 3) + 2
Parentheses first: 5(7) + 2
Multiply: 35 + 2
Answer: 37

Example 6:

Evaluate (x² + 5) ÷ 3 when x = 7

Substitute: (7² + 5) ÷ 3
Exponent: (49 + 5) ÷ 3
Parentheses: 54 ÷ 3
Answer: 18

Quick recap

Replace the variable with its value.
Then simplify using the order of operations.
Substitute variables with given numbers
Use parentheses when substituting (prevents errors)
Follow PEMDAS order of operations
Always show your steps
Check your arithmetic

✓ Quick check

If x = 3, what is 2x + 1?

2 × 3 + 1 = 7.

If y = 10, what is y − 4?

10 − 4 = 6.

Simple Equations

An equation says two expressions are equal. We solve it by finding the value of the variable, either by trial or by balancing (doing the same to both sides).

For x + 5 = 12, take 5 from both sides: x = 7. For 3x = 15, divide both sides by 3: x = 5.

Example 1: Solve x + 5 = 12.

x = 12 − 5 = 7.

Example 2: Solve 3x = 15.

x = 15 ÷ 3 = 5.

Quick recap

An equation balances two equal sides.
Do the same operation to both sides to solve.

✓ Quick check

Solve: x − 4 = 10.

x = 10 + 4 = 14.

Solve: 2x = 18.

x = 18 ÷ 2 = 9.