Algebra · Algebraic Expressions · Lesson 5

Simplifying Expressions Using the Properties of Operations

The commutative, associative and distributive properties are the rules that let us rearrange, regroup and remove brackets without changing the value of an expression. In this lesson you will use them to expand brackets such as $3 (x + 2)$ , simplify expressions that mix all four operations, and work confidently with fraction coefficients — skills you will need for solving equations and for the BECE.

NaCCA B7.2.2.1.5 ⏱ about 50 min

By the end you will be able to

✔State and use the commutative, associative and distributive properties with algebraic expressions.
✔Expand brackets, including those with negative and fraction multipliers.
✔Expand and simplify expressions such as $3 (x + 2) + 2 (x - 1)$ and $4 (2 a + b) - 3 (a - b)$ .
✔Simplify expressions involving all four operations in the correct order.

Key words

Commutative property: Changing the order does not change the answer for + and ×: $a + b = b + a$ , $ab = ba$ .
Associative property: Changing the grouping does not change the answer for + and ×: $(a + b) + c = a + (b + c)$ .
Distributive property: Multiplication spreads over addition and subtraction: $a (b + c) = ab + a c$ .
Expand: Remove brackets by multiplying every term inside by the term outside.

Curriculum reference

Content standard: B7.2.2.1 Simplify algebraic expressions involving the four basic operations and substituting values to evaluate algebraic expressions.

Indicator: Use properties of the four operations to simplify algebraic expressions with rational coefficients.

Starter

Two ways to pay for the same shopping

Akosua buys 3 "combos" at a provision shop in Accra. Each combo is one tin of milk costing GH₵x and one sachet of sugar costing GH₵2. The shopkeeper can add up one combo and multiply by 3: $3 (x + 2)$ . Or he can count 3 tins and 3 sachets: $3 x + 6$ . Both ways must give the same bill — so $3 (x + 2) = 3 x + 6$ . This is the distributive property, and it works for every price x.

1. The properties of the operations

You have used these rules with numbers since primary school. Because letters stand for numbers, the same rules work in algebra. Check each one with numbers in the table.

Property	In algebra	Check with numbers
Commutative (+)	$a + b = b + a$	3 + 5 = 5 + 3 = 8
Commutative (×)	$ab = ba$	4 × 6 = 6 × 4 = 24
Associative (+)	$(a + b) + c = a + (b + c)$	(2 + 3) + 4 = 2 + (3 + 4) = 9
Associative (×)	$(ab) c = a (b c)$	(2 × 3) × 4 = 2 × (3 × 4) = 24
Distributive	$a (b + c) = ab + a c$	3 × (10 + 2) = 30 + 6 = 36

2. Expanding brackets

To expand a bracket, multiply every term inside it by the term outside. The area model shows why: a rectangle 3 by $(x + 2)$ splits into a $3 \times x$ part and a $3 \times 2$ part.

▶ The rectangle 3 by (x + 2) is made of a 3x part and a 6 part, so 3(x + 2) = 3x + 6.

When the number outside is negative, multiply each term with its sign, using the sign rules.

▶ The −2 multiplies both terms. Negative × negative gives +8, not −8.

$5 (2 a - 1) = 10 a - 5$
$\frac{1}{2} (4 x + 6) = 2 x + 3$
$- (a - b) = - a + b$ (a minus sign in front of a bracket means multiply by −1)
$x (x + 3) = x^{2} + 3 x$

3. Expand, then collect like terms

Many BECE questions ask you to "expand and simplify". Do it in two clear stages: first remove all the brackets, then collect like terms.

1Expand each bracket: $4 (2 a + b) - 3 (a - b) = 8 a + 4 b - 3 a + 3 b$ . (Note $- 3 \times - b = + 3 b$ .)
2Collect the a-terms: $8 a - 3 a = 5 a$ .
3Collect the b-terms: $4 b + 3 b = 7 b$ .
4Answer: $5 a + 7 b$ .

4. Mixing all four operations

When an expression mixes operations, use BODMAS: work out brackets first, then multiplication and division, and only then add or subtract like terms.

Expression	Working	Simplest form
$\frac{7 x + 4 x - 2 x}{3}$	top: $9 x$ ; then $9 x \div 3$	$3 x$
$3 a \times 2 b + \frac{6 ab}{2}$	$6 ab + 3 ab$	$9 ab$
$(15 x^{2} \div 5 x) + (12 x \div 4)$	$3 x + 3 x$	$6 x$
$\frac{12 x + 8}{4}$	$\frac{12 x}{4} + \frac{8}{4}$	$3 x + 2$
$3 x^{2} y + 2 x y^{2} - 4 x^{2} y - 6 x y^{2}$	$x^{2} y$ terms: 3 − 4; $x y^{2}$ terms: 2 − 6	$- x^{2} y - 4 x y^{2}$

Worked examples

Try each one yourself first, then open the solution and compare every step.

Example 1

Expanding with positive and negative multipliers

Expand (a)

3 (x + 2)

(b)

- 2 (x - 4)

Show the solution

Step 1(a) $3 \times x + 3 \times 2 = 3 x + 6$ .
Step 2(b) $- 2 \times x = - 2 x$ and $- 2 \times (- 4) = + 8$ .
Step 3So $- 2 (x - 4) = - 2 x + 8$ .

Answer: (a) $3 x + 6$ (b) $- 2 x + 8$

Example 2

Expand and simplify

Simplify

3 (x + 2) + 2 (x - 1)

Show the solution

Step 1Expand: $3 x + 6 + 2 x - 2$ .
Step 2x-terms: $3 x + 2 x = 5 x$ .
Step 3Numbers: $6 - 2 = 4$ .
Step 4Answer: $5 x + 4$ . Check with $x = 1$ : $9 + 0 = 9$ and $5 + 4 = 9$ ✓

Answer: $5 x + 4$

Example 3

Subtracting a bracket

Simplify

4 (2 a + b) - 3 (a - b)

Show the solution

Step 1Expand: $8 a + 4 b - 3 a + 3 b$ (because $- 3 \times - b = + 3 b$ ).
Step 2a-terms: $8 a - 3 a = 5 a$ .
Step 3b-terms: $4 b + 3 b = 7 b$ .

Answer: $5 a + 7 b$

Example 4

Four operations together

Simplify (a)

\frac{7 x + 4 x - 2 x}{3}

(b)

(15 x^{2} \div 5 x) + (12 x \div 4)

Show the solution

Step 1(a) Top first: $7 x + 4 x - 2 x = 9 x$ . Then $\frac{9 x}{3} = 3 x$ .
Step 2(b) $15 x^{2} \div 5 x = 3 x$ and $12 x \div 4 = 3 x$ .
Step 3Add: $3 x + 3 x = 6 x$ .

Answer: (a) $3 x$ (b) $6 x$

Example 5

Terms with powers

Simplify

3 x^{2} y + 2 x y^{2} - 4 x^{2} y - 6 x y^{2}

Show the solution

Step 1 $x^{2} y$ and $x y^{2}$ are different (unlike) terms.
Step 2 $x^{2} y$ terms: $3 - 4 = - 1$ , giving $- x^{2} y$ .
Step 3 $x y^{2}$ terms: $2 - 6 = - 4$ , giving $- 4 x y^{2}$ .

Answer: $- x^{2} y - 4 x y^{2}$

Summary

Commutative: $a + b = b + a$ and $ab = ba$ . Associative: grouping does not matter for + and ×. Neither works for − or ÷.
Distributive: $a (b + c) = ab + a c$ . Multiply EVERY term in the bracket: $3 (x + 2) = 3 x + 6$ , not $3 x + 2$ .
With a negative multiplier, use the sign rules: $- 2 (x - 4) = - 2 x + 8$ .
Expand and simplify in two stages: remove brackets, then collect like terms.
With mixed operations, do × and ÷ before + and −; divide every term on top: $\frac{12 x + 8}{4} = 3 x + 2$ .

Two ways to pay for the same shopping

1. The properties of the operations

2. Expanding brackets

3. Expand, then collect like terms

4. Mixing all four operations

Worked examples

Expanding with positive and negative multipliers

Expand and simplify

Subtracting a bracket

Four operations together

Terms with powers

Summary

Practice: 15 questions