Handro Learning

Algebra is a part of mathematics in which letters are used to represent numbers. This lets us write rules, patterns, formulas and expressions without knowing every value at the start — for example, instead of saying "multiply a number by 3 and add 5," we can write 3x + 5. You will use algebra throughout CSEC Mathematics, so it is important to understand the basic language of algebra and how to substitute numbers correctly into algebraic expressions.

What You Will Learn

Identify variables, constants, coefficients, terms, and algebraic expressions
Understand common algebraic notation
Recognize multiplication, division, and powers in algebra
Substitute numbers correctly for variables
Use brackets when substituting negative numbers
Apply BODMAS when evaluating expressions
Evaluate expressions containing powers, brackets, fractions, and more than one variable
Avoid common substitution mistakes

What Is Algebra?

Algebra uses letters and symbols to represent numerical values.

For example:

x + 5

The letter x represents a number.

If one exercise book costs x dollars, then the cost of 4 exercise books can be written as:

This means:

4 × x

Key Vocabulary

Word	Meaning	Example
Variable	A letter that represents an unknown or changing value	In 3x + 5, x is the variable
Constant	A fixed number that does not change	In 3x + 5, 5 is the constant
Coefficient	The number multiplying a variable	In 3x, the coefficient is 3
Term	A single number, variable, or product of numbers and variables	In 3x + 5, the terms are 3x and 5
Expression	A mathematical phrase without an equals sign	4x − 7
Substitution	Replacing a variable with a given number	If x = 4, then 3x + 5 = 3(4) + 5

Good to know: Constants can be any fixed value, including negatives, e.g. 5, −12, 20. A coefficient can also be negative or implied — for example, the coefficient of y is −1, because −y = −1y.

What Is a Term?

A term is a number, a variable, or a product of numbers and variables. Terms in an expression are separated by addition or subtraction signs.

Example

3x + 2y − 5

This expression has three terms:

3x
2y
−5

What Is an Algebraic Expression?

An algebraic expression is made up of numbers, variables, and mathematical operations. It does not contain an equals sign.

Examples:

3x + 5

2a − 4b

x² + 3y − 7

Remember: An expression such as 3x + 5 has no equals sign. When an equals sign is included, such as 3x + 5 = 20, it is an equation.

Terms in Algebra

Expression	Terms	Coefficient(s)	Constant
3x + 5	3x, 5	3	5
4y − 7	4y, −7	4	−7
2a² − b	2a², −b	2, −1	None
5(y + 1)	One grouped expression	5 outside the bracket	Depends on expansion

Note: The coefficient of x is 1, because x = 1x. The coefficient of −x is −1, because −x = −1x.

How Algebraic Notation Works

In algebra, some operations are written differently from ordinary arithmetic.

Operation	Arithmetic Meaning	Algebraic Notation
Multiplication	4 multiplied by x	4x
Product of variables	a multiplied by b	ab
Repeated multiplication	x × x × x	x³
Division	y divided by 3	y3

Multiplication in Algebra

When a number is multiplied by a variable, the multiplication sign is usually left out.

4 × x = 4x

6 × y = 6y

When two variables are multiplied:

a × b = ab

Important: 4x means 4 × x. It does not mean the number 4 placed beside another digit.

Powers in Algebra

Repeated multiplication can be written using powers.

x × x = x²

x × x × x = x³

Therefore:

3x² = 3 × x × x

What Is Substitution?

Substitution means replacing a variable with its given numerical value.

Suppose:

x = 4

Evaluate:

x + 6

Replace x with 4:

4 + 6

= 10

Answer: 10

Steps for Substitution

Replace each variable with its given numerical value. Use brackets around the substituted value.
Evaluate powers before multiplication, division, addition, or subtraction.
Multiply or divide from left to right.
Add or subtract from left to right to obtain the final answer.

Worked Example 1: Basic Substitution

Evaluate 3x + 4 when x = 5.

Substitute 5 for x:

3(5) + 4

Multiply:

15 + 4

= 19

Answer: 19

Worked Example 2: More Than One Variable

Evaluate 4x + 3y when x = 3 and y = −2.

Substitute:

4(3) + 3(−2)

Multiply:

12 − 6

= 6

Answer: 6

Negative Numbers Need Brackets

When substituting a negative value, put the entire value inside brackets.

Evaluate a² − 2a when a = −3.

Substitute:

(−3)² − 2(−3)

Evaluate the power:

9 − 2(−3)

Multiply:

9 − (−6)

Subtracting a negative is the same as adding:

9 + 6

= 15

Answer: 15

Be Careful with Negative Numbers and Powers

These two expressions do not mean the same thing:

(−3)² = 9

because:

(−3)(−3) = 9

But:

−3² = −9

In the second expression, only the 3 is squared. The negative sign is outside the power.

CSEC Tip: When substituting a negative number for a variable, use brackets. If x = −3, then x² should be written as (−3)².

Worked Example 3: Squares and Several Variables

Evaluate 3p²q when p = −2 and q = 3.

Substitute:

3(−2)²(3)

Evaluate the power:

3(4)(3)

Multiply:

12(3)

= 36

Answer: 36

Worked Example 4: Brackets and Powers

Evaluate 2(x + y)² − y when x = −3 and y = 4.

Substitute:

2(−3 + 4)² − 4

Work inside the brackets:

2(1)² − 4

Evaluate the power:

2(1) − 4

Multiply:

2 − 4

= −2

Answer: −2

Substitution with a Fraction

Some substitution questions contain a fraction. Treat the numerator and denominator as separate groups before dividing.

Evaluate p² − 3qr + q when p = −4, q = 2, and r = 3.

Substitute:

(−4)² − 3(2)3 + 2

Evaluate the power and multiplication:

16 − 65

Simplify the numerator:

105

= 2

Answer: 2

Another Fraction Example

Evaluate x + yz + 3 when x = 4, y = 2, and z = −1.

Substitute:

4 + 2−1 + 3

Simplify the numerator and denominator:

= 3

Answer: 3

Double Negatives

A common CSEC substitution problem occurs when a negative number is being subtracted.

Evaluate m − n when m = 5 and n = −4.

Substitute:

5 − (−4)

Subtracting a negative becomes addition:

5 + 4

= 9

Answer: 9

Using BODMAS

BODMAS helps us remember the correct order of operations:

B — Brackets
O — Orders or powers
D — Division
M — Multiplication
A — Addition
S — Subtraction

Multiplication and division have the same priority and are performed from left to right.

Addition and subtraction also have the same priority and are performed from left to right.

Worked Example 5: Full BODMAS

Evaluate 2x² + 3x − 4 when x = 3.

Substitute:

2(3)² + 3(3) − 4

Evaluate the power:

2(9) + 3(3) − 4

Multiply:

18 + 9 − 4

Add and subtract from left to right:

27 − 4

= 23

Answer: 23

Substitution into a Formula

The same substitution method is used when working with formulas.

The perimeter of a rectangle is:

P = 2l + 2w

Find the perimeter when l = 8 cm and w = 5 cm.

Substitute:

P = 2(8) + 2(5)

Multiply:

P = 16 + 10

P = 26 cm

Answer: 26 cm

Common Student Errors

Error 1: Forgetting That Letters Beside Numbers Mean Multiplication

If x = 5, then:

3x = 3(5) = 15

It does not mean 35.

Error 2: Dropping the Negative Sign

If x = −2, then:

4x = 4(−2) = −8

The negative sign is part of the value and must be included.

Error 3: Not Using Brackets with Negative Values

If:

x = −4

then:

x² = (−4)² = 16

Writing the brackets clearly shows that the whole value −4 is being squared.

Error 4: Ignoring BODMAS

For:

3(4) + 5

multiply before adding:

12 + 5 = 17

Error 5: Mishandling a Double Negative

For:

5 − (−4)

subtracting a negative becomes addition:

5 + 4 = 9

Practice Questions

Evaluate 4x + 3y when x = 3 and y = −2.
Evaluate a² − 2a when a = −3.
Evaluate m − n when m = 5 and n = −4.
Evaluate 3p²q when p = −2 and q = 3.
Evaluate x + yz + 3 when x = 4, y = 2, and z = −1.
Evaluate u² − v² when u = −5 and v = −2.
Evaluate 2(a − b)² when a = 3 and b = 7.
Evaluate (3r + s)³ when r = −1 and s = 4.
Evaluate xyz when x = 6, y = −3, and z = 2.
Evaluate a³ − b² when a = 2 and b = −3.

Answers

6
4(3) + 3(−2) = 12 − 6 = 6
15
(−3)² − 2(−3) = 9 + 6 = 15
9
5 − (−4) = 5 + 4 = 9
36
3(−2)²(3) = 3(4)(3) = 36
3
4 + 2−1 + 3 = 62 = 3
21
(−5)² − (−2)² = 25 − 4 = 21
32
2(3 − 7)² = 2(−4)² = 2(16) = 32
1
[3(−1) + 4]³ = (1)³ = 1
−9
6(−3)2 = −182 = −9
−1
(2)³ − (−3)² = 8 − 9 = −1

Lesson Summary

Algebra uses letters to represent numbers.
A variable is a letter used to represent a numerical value.
A constant is a fixed numerical value.
A coefficient is the numerical multiplier of a variable.
An algebraic expression contains terms joined by mathematical operations and does not contain an equals sign.
Writing 4x means 4 × x. Writing ab means a × b. Writing x³ means x × x × x.
Substitution means replacing a variable with its given numerical value.
Always place negative substituted values inside brackets.
Remember that (−3)² = 9 but −3² = −9.
Use BODMAS to determine the correct order of calculation.
For fractions, simplify the numerator and denominator before performing the final division.

Introduction to Algebra and Substitution

Video Lesson

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Written Lesson

What You Will Learn

What Is Algebra?

Key Vocabulary

What Is a Term?

What Is an Algebraic Expression?

Terms in Algebra

How Algebraic Notation Works

Multiplication in Algebra

Powers in Algebra

What Is Substitution?

Steps for Substitution

Worked Example 1: Basic Substitution

Worked Example 2: More Than One Variable

Negative Numbers Need Brackets

Be Careful with Negative Numbers and Powers

Worked Example 3: Squares and Several Variables

Worked Example 4: Brackets and Powers

Substitution with a Fraction

Another Fraction Example

Double Negatives

Using BODMAS

Worked Example 5: Full BODMAS

Substitution into a Formula

Common Student Errors

Practice Questions

Answers

Lesson Summary

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