Transcript 1-7 Simplifying Expressions

1-7 Simplifying Expressions
Preview
Warm Up
California Standards
Lesson Presentation
1-7 Simplifying Expressions
Warm Up
Evaluate.
1. 42
3. –23
16
–8
2. |5 – 16| 11
4. |3 – 7|
4
Translate each word phrase into a numerical
or algebraic expression.
5. The product of 8 and 6 8  6
6. The difference of 10y and 4 10y – 4
Simplify each fraction.
7.
8
8.
1-7 Simplifying Expressions
California
Standards
1.0 Students use properties of numbers to
demonstrate whether assertions are true or
false.
25.1 Students use properties of numbers to
construct simple, valid arguments (direct and
indirect) for, or formulate counterexamples
to, claimed assertions.
1-7 Simplifying Expressions
Vocabulary
order of operations
terms
like terms
coefficient
1-7 Simplifying Expressions
When an expression contains more than one
operation, the order of operations tells you
which operation to perform first.
1-7 Simplifying Expressions
Order of Operations
First:
Perform operations inside grouping symbols.
Second: Evaluate powers.
Third:
Perform multiplication and division from left to right.
Fourth:
Perform addition and subtraction from left to right.
1-7 Simplifying Expressions
Grouping symbols include parentheses ( ),
brackets [ ], and braces { }. If an expression
contains more than one set of grouping symbols,
begin with the innermost set. Follow the order of
operations within that set of grouping symbols
and then work outward.
1-7 Simplifying Expressions
Helpful Hint
Fraction bars, radical symbols, and absolute-value
symbols can also be used as grouping symbols.
Remember that a fraction bar indicates division.
1-7 Simplifying Expressions
Additional Example 1: Simplifying Numerical
Expressions
Simplify each expression.
A. 15 – 2  3 + 1
There are no grouping symbols.
15 – 2  3 + 1
Multiply.
15 – 6 + 1
Subtract.
9+1
10
B. 12 + 32 + 10 ÷ 2
12 + 32 + 10 ÷ 2
12 + 9 + 10 ÷ 2
12 + 9 + 5
26
Add.
There are no grouping symbols.
Evaluate powers. The exponent
applies only to the 3.
Divide.
Add.
1-7 Simplifying Expressions
Additional Example 1: Simplifying Numerical
Expressions
Simplify each expression.
C.
The fraction bar is a grouping
symbol.
Evaluate powers. The exponent
applies only to the 4.
Multiply above the bar and
subtract below the bar.
Add above the bar and then
divide.
1-7 Simplifying Expressions
Check It Out! Example 1a
Simplify the expression.
There are no grouping symbols.
Rewrite division as multiplication.
Multiply.
48
1-7 Simplifying Expressions
Check It Out! Example 1b
Simplify the expression.
The square root sign acts as a
grouping symbol.
Subtract.
37
Take the square root.
21
Multiply.
1-7 Simplifying Expressions
Check It Out! Example 1c
Simplify the expression.
The division bar acts as a grouping
symbol.
Add and evaluate the power.
Multiply, subtract and simplify.
1-7 Simplifying Expressions
Additional Example 2: Retail Application
A shop offers gift-wrapping services at three
price levels. The amount of money collected
for wrapping gifts on a given day can be
found using the expression 2B + 4S + 7D. On
Friday the shop wrapped 10 basic packages B,
6 super packages S, and 5 deluxe packages D.
Use the expression to find the amount of
money collected for gift-wrapping on Friday.
2B + 4S +7D
2(10) + 4(6) + 7(5) Substitute values for variables.
Multiply.
20 + 24 + 35
Add.
79
A total of $79 was collected on Friday.
1-7 Simplifying Expressions
Check It Out! Example 2
A formula for a player’s total number of bases
is Hits + D + 2T + 3H. Use this expression to
find Hank Aaron’s total bases for 1959, when
he had 223 hits, 46 doubles, 7 triples, and 39
home runs.
Hits + D + 2T + 3H
223 + 46 + 2(7) + 3(39)Substitute values for variables.
223 + 46 + 14 + 117
Multiply.
400
Add.
Hank Aaron’s total number of bases for 1959 was 400.
1-7 Simplifying Expressions
The terms of an expression are the parts to be
added or subtracted. Like terms are terms that
contain the same variables raised to the same
powers. Constants are also like terms.
Like terms
Constant
4x – 3x + 2
1-7 Simplifying Expressions
A coefficient is a number multiplied by a variable.
Like terms can have different coefficients. A
variable written without a coefficient has a
coefficient of 1.
Coefficients
1x2 + 3x
1-7 Simplifying Expressions
Like terms can be combined. To combine like
terms, use the Distributive Property.
Distributive Property
ax – bx = (a – b)x
Example
7x – 4x = (7 – 4)x
= 3x
Notice that you can combine like terms by
adding or subtracting the coefficients. Keep the
variables and exponents the same.
1-7 Simplifying Expressions
Additional Example 3: Combining Like Terms
Simplify the expression by combining like
terms.
A. 72p – 25p
72p – 25p
47p
72p and 25p are like terms.
Subtract the coefficients.
1-7 Simplifying Expressions
Additional Example 3: Combining Like Terms
Simplify the expression by combining like
terms.
B.
A variable without a coefficient
has a coefficient of 1.
and
are like terms.
Write 1 as .
Add the coefficients.
1-7 Simplifying Expressions
Additional Example 3: Combining Like Terms
Simplify the expression by combining like
terms.
C. 0.5m + 2.5n
0.5m + 2.5n
0.5m and 2.5n are not like terms.
0.5m + 2.5n
Do not combine the terms.
1-7 Simplifying Expressions
Caution!
Add or subtract only the coefficients.
6.8y² – y² ≠ 6.8
1-7 Simplifying Expressions
Check It Out! Example 3
Simplify by combining like terms.
a. 16p + 84p
16p + 84p
100p
b. –20t – 8.5t
–20t – 8.5t
–28.5t
c. 3m2 + m3 – m2
3m2 – m2 + m3
2m2 + m3
16p + 84p are like terms.
Add the coefficients.
20t and 8.5t are like terms.
Subtract the coefficients.
3m2 and – m2 are like terms.
Subtract coefficients.
1-7 Simplifying Expressions
Additional Example 4: Simplifying Algebraic
Expressions
Use properties and operations to show that
14x + 4(2 + x) simplifies to 18x + 8.
Reasons
1.
Statements
14x + 4(2 + x)
2.
3.
14x + 4(2) + 4(x)
14x + 8 + 4x
Distributive Property
4.
14x + 4x + 8
5.
(14x + 4x) + 8
6.
18x + 8
Multiply.
Commutative Property of
Addition
Associative Property of
Addition
Combine like terms.
1-7 Simplifying Expressions
Check It Out! Example 4
Use properties and operations to show that
6(x – 4) + 9 simplifies to 6x – 15.
Statements
1.
2.
3.
4.
Reasons
6(x – 4) + 9
6x – 6(4) + 9
6x – 24 + 9
Distributive Property
6x – 15
Combine like terms.
Multiply.
1-7 Simplifying Expressions
Lesson Quiz
Simplify each expression.
1. 165 + 27 + 3 + 5 200
2.
8
3. The volume of a storage box can be found using
the expression lw(w + 2). Find the volume of
the box if l = 3 feet and w = 2 feet. 24 ft3
Simplify each expression by combining like
terms.
4.
5. 14c2 – 9c
14c2 – 9c
6. Use properties and operations to show that
24a + b² + 3a + 2b² simplifies to 27a + 3b².
Check students’ work.