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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.

\[(7x - 3)5 = 5(7x - 3)\]

Apply the distributive law.

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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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