Algebra & Sequences

Algebra might sound complicated, but it is really just using letters to stand for numbers you do not know yet. In the GL 11+ exam, you will need to understand simple expressions, solve equations, and work with number sequences. This lesson teaches you how.

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What Is Algebra?

In algebra, we use letters (called variables) to represent unknown numbers.

Expression	Meaning
a + 5	A number plus 5
3b	3 times a number
c - 7	A number minus 7
d/4	A number divided by 4
2n + 1	Double a number, then add 1

Key Vocabulary

• Expression: A collection of terms (e.g. 3a + 2b - 5). No equals sign.
• Equation: An expression with an equals sign (e.g. 3a + 2 = 17).
• Variable: A letter representing an unknown number.
• Coefficient: The number in front of a variable (in 5x, the coefficient is 5).
• Term: A single part of an expression (in 3a + 2b - 5, there are three terms: 3a, 2b, and -5).

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Simplifying Expressions

Collecting Like Terms

Like terms have the same letter (or no letter at all). You can add or subtract like terms.

Simplify: 3a + 5b + 2a - b

• Collect the "a" terms: 3a + 2a = 5a
• Collect the "b" terms: 5b - b = 4b
• Answer: 5a + 4b

Simplify: 7x + 3 - 2x + 9

• x terms: 7x - 2x = 5x
• Number terms: 3 + 9 = 12
• Answer: 5x + 12

Remember: You cannot combine different letters. 3a + 2b cannot be simplified further.

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Substitution

Substitution means replacing a letter with a number and working out the answer.

If a = 4 and b = 3, find the value of 2a + 3b - 1

• 2 × 4 + 3 × 3 - 1
• = 8 + 9 - 1
• = 16

If x = 5, find the value of x² + 2x

• 5² + 2 × 5
• = 25 + 10
• = 35

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Solving Equations

An equation contains an equals sign. To solve it, find the value of the unknown letter.

The Balance Method

Think of the equation as a set of scales. Whatever you do to one side, you must do to the other to keep it balanced.

Solve: x + 7 = 15

• Subtract 7 from both sides: x = 15 - 7 = 8

Solve: 3x = 24

• Divide both sides by 3: x = 24 ÷ 3 = 8

Solve: 2x + 5 = 19

• Subtract 5: 2x = 14
• Divide by 2: x = 7

Solve: 4x - 3 = 17

• Add 3: 4x = 20
• Divide by 4: x = 5

Two-Step Equations — Worked Example

Solve: 5x + 8 = 33

Step 1: Subtract 8 from both sides: 5x = 25

Step 2: Divide both sides by 5: x = 5

Check: 5 × 5 + 8 = 25 + 8 = 33 ✓

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Forming Equations from Word Problems

Tom thinks of a number, doubles it, and adds 9. His answer is 31. What number did Tom think of?

• Let the number be n.
• 2n + 9 = 31
• 2n = 22
• n = 11

A rectangle has a width of 5 cm and a length of (x + 3) cm. Its perimeter is 34 cm. Find x.