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# Using laws to simplify expressions

## 6.5 Using laws to simplify expressions

### Distributive law

To “distribute” means to deliver or pass around. For example, before a test, the teacher must distribute the test paper to each student in the class. Recall the distributive property of multiplication over addition (or subtraction):

We also call the distributive property the distributive law.

For example, we use the distributive law to multiply out the following:

$5(y - 6)$

We can’t simplify $$5y - 30$$ any further, because $$5y$$ and $$- 30$$ are not like terms.

## Worked example 6.17: Distributive law

Distribute in order to expand this expression: $$5(10 + g)$$

### Apply the distributive law:

#### Commutative law

Remember the commutative property of multiplication:

$a \times b = b \times a$

For example, $$3 \times 5 = 5 \times 3$$.

We also call the commutative property the commutative law.

## Worked example 6.18: Distributive and commutative law

Expand the following expression by multiplying out the brackets:

$(7x - 3)5$

### Use the commutative law.

$(7x - 3)5 = 5(7x - 3)$

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## Worked example 6.19: Simplify trinomials

Distribute in order to expand this expression:

$5(4f^{2} + 2f + 5)$

### Apply the distributive law.

For the expression $$5(4f^{2} + 2f + 5)$$, you must distribute the 5 to all three of the terms in the brackets.

Remember, to simplify an expression:

• separate it into terms
• simplify each term (if needed)
• add or subtract like terms from left to right.

## Worked example 6.20: Simplify expressions with like terms

Simplify the following expression:

$5 + 2(5n - 7)$

### Separate the expression into terms.

$$5 + 2(5n - 7)$$ has two terms: $$5$$ and $$+ 2(5n - 7)$$.

### Simplify using the distributive law.

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#### Adding and subtracting algebraic terms

We have already come across the commutative and associative laws of operations. We will now use these laws to help us form equivalent algebraic expressions.

• Commutative law: The order in which we add or multiply numbers does not change the answer:
$$a + b = b + a$$ and $$ab = ba$$.
• Associative law: The way in which we group three or more numbers when adding or multiplying does not change the answer:
$$(a + b) + c = a + (b + c)$$ and $$(ab)c = a(bc)$$.

## Worked example 6.21: Adding algebraic terms

Add $$5r^{2} + 5r - 4$$ to $$4r^{2} - r$$.

### Write and simplify the expression.

We must start with $$4r^{2} - r$$ and add $$5r^{2} + 5r - 4$$ to it.

$\left( 4r^{2} - r \right) + \left( 5r^{2} + 5r - 4 \right)$ $= 4r^{2} - r + 5r^{2} + 5r - 4$ $= 9r^{2} + 4r - 4$
$+ \ (5r^{2} + 5r - 4)\ = + 1(5r^{2} + 5r - 4)$

When we distribute in $$+1$$, the signs don’t change.

## Worked example 6.22: Subtracting algebraic terms

Subtract $$4r - t$$ from $$5r + 5t - 4$$.

### Write and simplify the expression.

We must start with $$5r + 5t - 4$$ and subtract $$4r - t$$ from it.

$(5r + 5t - 4) - (4r - t)$ $= 5r + 5t - 4 - 4r + t$ $= r + 6t - 4$
$- (4r - t)\ = - 1(4r - t)$

When we distribute in the $$−1$$, all the signs change!

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