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Page 1: Simplifying expressions

Simplifying Expressions

By Zain Bin Masood

Senior Maths Teacher

At The Intellect School

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Objective

This presentation is designed to give a brief review of simplifying algebraic expressions and evaluating algebraic expressions.

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Algebraic Expressions

An algebraic expression is a collection of real numbers, variables, grouping symbols and operation symbols.

Here are some examples of algebraic expressions.

27,7

5

3

1,4,75 2 xxyxx

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Consider the example:

The terms of the expression are separated by addition. There are 3 terms in this example and they are

.

The coefficient of a variable term is the real number factor. The first term has coefficient of 5. The second term has an unwritten coefficient of 1.

The last term , -7, is called a constant since there is no variable in the term.

75 2 xx

7,,5 2 xx

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Let’s begin with a review of two important skills for simplifying expression, using the Distributive Property and combining like terms. Then we will use both skills in the same simplifying problem.

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Distributive Property

a ( b + c ) = ba + ca

To simplify some expressions we may need to use the Distributive Property

Do you remember it?

Distributive Property

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Examples

Example 1: 6(x + 2)

Distribute the 6.

6 (x + 2) = x(6) + 2(6)

= 6x + 12

Example 2: -4(x – 3)

Distribute the –4.

-4 (x – 3) = x(-4) –3(-4)

= -4x + 12

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Practice Problem

Try the Distributive Property on -7 ( x – 2 ) .

Be sure to multiply each term by a –7.

-7 ( x – 2 ) = x(-7) – 2(-7)

= -7x + 14

Notice when a negative is distributed all the signs of the terms in the ( )’s change.

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Examples with 1 and –1.

Example 3: (x – 2)

= 1( x – 2 )

= x(1) – 2(1)

= x - 2

Notice multiplying by a 1 does nothing to the expression in the ( )’s.

Example 4: -(4x – 3)

= -1(4x – 3)

= 4x(-1) – 3(-1)

= -4x + 3

Notice that multiplying by a –1 changes the signs of each term in the ( )’s.

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Like Terms

Like terms are terms with the same variables raised to the same power.

Hint: The idea is that the variable part of the terms must be identical for them to be like terms.

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Examples

Like Terms

5x , -14x

-6.7xy , 02xy

The variable factors are

identical.

Unlike Terms

5x , 8y

The variable factors are

not identical.

22 8,3 xyyx

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Combining Like Terms

Recall the Distributive Property

a (b + c) = b(a) +c(a)

To see how like terms are combined use the

Distributive Property in reverse.

5x + 7x = x (5 + 7)

= x (12)

= 12x

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Example

All that work is not necessary every time.

Simply identify the like terms and add their

coefficients.

4x + 7y – x + 5y = 4x – x + 7y +5y

= 3x + 12y

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Collecting Like Terms Example

31316

terms.likeCombine

31334124

terms.theReorder

33124134

2

22

22

yxx

yxxxx

xxxyx

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Both Skills

This example requires both the Distributive

Property and combining like terms.

5(x – 2) –3(2x – 7)

Distribute the 5 and the –3.

x(5) - 2(5) + 2x(-3) - 7(-3)

5x – 10 – 6x + 21

Combine like terms.

- x+11

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Simplifying Example

431062

1 xx

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Simplifying Example

Distribute. 43106

2

1 xx

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Simplifying Example

Distribute. 43106

2

1 xx

12353

3432

110

2

16

xx

xx

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Simplifying Example

Distribute.

Combine like terms.

431062

1 xx

12353

3432

110

2

16

xx

xx

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Simplifying Example

Distribute.

Combine like terms.

431062

1 xx

12353

3432

110

2

16

xx

xx

76 x

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Evaluating Expressions

Remember to use correct order of operations.

Evaluate the expression 2x – 3xy +4y when

x = 3 and y = -5.

To find the numerical value of the expression, simply replace the variables in the expression with the appropriate number.

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Example

Evaluate 2x–3xy +4y when x = 3 and y = -5.

Substitute in the numbers.

2(3) – 3(3)(-5) + 4(-5)

Use correct order of operations.

6 + 45 – 20

51 – 20

31

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Evaluating Example

1and2when34Evaluate 22 yxyxyx

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Evaluating Example

Substitute in the numbers.

1and2when34Evaluate 22 yxyxyx

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Evaluating Example

Substitute in the numbers.

1and2when34Evaluate 22 yxyxyx

22 131242

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Evaluating Example

Remember correct order of operations.

1and2when34Evaluate 22 yxyxyx

22 131242

Substitute in the numbers.

131244

384

15

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Common Mistakes

Incorrect Correct

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