Simplifying expressions

Simplifying Expressions

By Zain Bin Masood

Senior Maths Teacher

At The Intellect School

Objective

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

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

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

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.

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

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

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.

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.

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.

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

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

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

Collecting Like Terms Example

31316

terms.likeCombine

31334124

terms.theReorder

33124134

2

22

22

yxx

yxxxx

xxxyx

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

Simplifying Example

431062

1 xx

Simplifying Example

Distribute. 43106

2

1 xx

Simplifying Example

Distribute. 43106

2

1 xx

12353

3432

110

2

16

xx

xx

Simplifying Example

Distribute.

Combine like terms.

431062

1 xx

12353

3432

110

2

16

xx

xx

Simplifying Example

Distribute.

Combine like terms.

431062

1 xx

12353

3432

110

2

16

xx

xx

76 x

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.

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

Evaluating Example

1and2when34Evaluate 22 yxyxyx

Evaluating Example

Substitute in the numbers.

1and2when34Evaluate 22 yxyxyx

Evaluating Example

Substitute in the numbers.

1and2when34Evaluate 22 yxyxyx

22 131242

Evaluating Example

Remember correct order of operations.

1and2when34Evaluate 22 yxyxyx

22 131242

Substitute in the numbers.

131244

384

15

Common Mistakes

Incorrect Correct