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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