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Evaluating Expressions and Combining Like Terms

Evaluating Expressions• Vocabulary:

– Variable – A symbol, usually a letter of the alphabet, such as the letter n, that is used to represent a number.

– Variable expression (A.K.A. - Algebraic Expression) – An expression, such as n – 5, that consists of one or more numbers and variables along with one or more arithmetic operations. (Note: No equal sign)

– Evaluate a Variable Expression – write the expression, substitute a number for each variable, and simplify the result.

How do you describe a variable expression?

Variable Expression

Meaning Operation

5x, 5·x, (5)(x) (same as x·5)

5 times x Multiplication

5 divided by x

Division

5 + x (same as x + 5) 5 plus x Addition

5 – x 5 minus x subtraction

xx

÷5,5

Substitute 4 for n. Simplify

Simplify (means to solve the problem or perform as many of the indicated operations as possible.)

Solution: Substitute 4 for n. Simplify

Solution:

Evaluate a Variable Expression

• Example 1: Evaluate each expression when n = 4.

a. n + 3 n + 3 = 4 + 3 = 7 b. n – 3 n – 3 = 4 – 3 = 1

Substitute 8 for x. Simplify

Solution:

Solution:

Using parenthesis is the preferred method to show multiplication. Additional ways to show multiplication are 5 · 8 and 5 x 8.

Substitute 8 for x. Simplify

Recall that division problems are also fractions – this problem could be written as:

44

2; 48

4

xx

because

x

=÷

=

=

Evaluate an Algebraic Expression

• Example 2: Evaluate each expression if x = 8. a. 5x 5x = 5(8) = 40 b. x ÷ 4 x ÷ 4 = 8 ÷ 4 = 2

Substitute 4 for x; 6 for y. simplify

Solution:

Evaluating More Expressions

• Example 3: Evaluate each expression if x = 4, y = 6, and z = 24.

a. 5xy 5xy = 5(4)(6) = 120 b.

= 4

Solution:

yz

624

=yz

Substitute 24 for z; 6 for y. Simplify.

A

A

A

A

A

A

Now You Try… Evaluate each expression given that a = 6, b = 12,

and c = 3.

1. 4ac 2. a ÷ c 3. a + b + c 4. ba 5. b – c 6. c ÷ b

Substitute the value for a = 6 and c = 3 into the problem and multiply

Click to return to “You try it” slide

You Try #1Evaluate each expression given that a = 6,

b = 12, and c = 3. 1. 4ac 4ac = 4(6)(3) = (24)(3) = 72

Substitute the value for a = 6 and c = 3 into the problem and divide

Click to return to “You try it” slide

You Try #2Evaluate each expression given that a =

6, b = 12, and c = 3. 2. a ÷ c a ÷ c = 6 ÷ 3 = 2

Substitute the value for a = 6, b=12, and c = 3 into the problem, then add.

Click to return to “You try it” slide

You Try #3Evaluate each expression given that a = 6,

b = 12, and c = 3. 3. a + b + c a + b + c = 6 + 12 + 3 = 18 + 3 = 21

Substitute the value for b=12 and a = 6 into the problem, then multiply.

Click to return to “You try it” slide

You Try #4Evaluate each expression given that a =

6, b = 12, and c = 3. 4. ba ba = (12)(6) = 72

Substitute the value for b=12 and a = 3 into the problem, then subtract.

Click to return to “You try it” slide

You Try #5Evaluate each expression given that a =

6, b = 12, and c = 3. 5. b - c b – c = 12 – 3 = 9

Substitute the value for c=3 and b = 12 into the problem, then Divide

Note: It is better to rewrite this division problem as a fraction. This fraction can now be reduced to its simplest form.

Divide both numerator and denominator by the GCF = (3) to reduce this fraction.

It is OK to have a fraction as an answer.

Click to return to “You try it” slide

You Try #6Evaluate each expression given that a =

6, b = 12, and c = 3. 6. c ÷ b

123

==÷bcbc

41

33

123

=÷

÷

Combining Like Terms• Now that we have seen some algebraic

expressions, we need to know how to simplify them.

• Vocabulary – Like terms: In an expression, like terms are the

terms that have the same variables, raised to the same powers (same exponents).

• i.e. 4x and -3x or 2y2 and –y2 – Coefficient: A constant that multiplies a variable.

• i.e. the 3 in 3a or the -1 in –b

Combining Like Terms• In algebra we often get very long

expressions, which we need to make simpler. Simpler expressions are easier to solve!

• To simplify an expression we collect like terms. Like terms include letters that are the same and numbers.

Let’s try one…• Step One: Write the expression. 4x + 5x -2 - 2x + 7 • Collect all the terms together which are alike. Remember that each term

comes with an operation (+,-) which goes before it. 4x, 5x, and -2x -2 and 7 • Simplify the variable terms. 4x+5x-2x = 9x-2x = 7x • Simplify the constant (number) terms. -2+7 = 5 • You have a simplified expression by writing all of the results from

simplifying. 7x + 5

Another example…• 10x – 4y + 3x2 + 2x – 2y 3x2 10x, 2x -4y – 2y • 3x2 + 12x – 6y

Remember you cannot combine terms with

the same variable but different exponents.

Now you try…Simplify the following: • 5x + 3y - 6x + 4y + 3z • 3b - 3a - 5c + 4b • 4ab – 2a2b + 5 – ab + ab2 + 2a2b + 4 • 5xy – 2yx + 7y + 3x – 4xy + 2x

A

A

A

A

You Try #1• Simplify the following: 1. 5x + 3y - 6x + 4y + 3z 5x, -6x 3y, 4y 3z -x + 7y + 3z

You Try #2• Simplify the following: 2. 3b - 3a - 5c + 4b 3b, 4b -3a -5c -3a + 7b – 5c

You Try #3• Simplify the following: 3. 4ab – 2a2b + 5 – ab + ab2 + 2a2b + 4 4ab, -ab -2a2b, 2a2b 5, 4 ab2 3ab + ab2 + 9

You Try #4• Simplify the following: 4. 5xy – 2yx + 7y + 3x – 4xy + 2x 5xy, -2yx, -4xy 7y 3x, 2x -xy + 7y + 5x

Conclusion• A variable or algebraic expression is an

expression that consists of one or more ________ and _________ along with one or more ________ _________. (Note: No _______ sign)

• To evaluate an expression write the _________, substitute a _______ for each variable, and _________ the result.

numbers variables

arithmetic operationsequal

expression numbersimplify

Conclusion Continued…• In an expression, __________ are

the terms that have the same ________, raised to the same ________ (same exponents).

• A coefficient is a number that ________ a variable.

like terms

variables

power

multiplies