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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.
Vocabulary
order of operations
terms
like terms
coefficient
When an expression contains more than one
operation, the order of operations tells you
which operation to perform first.
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.
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.
Helpful Hint
Fraction bars, radical symbols, and absolute-value
symbols can also be used as grouping symbols.
Remember that a fraction bar indicates division.
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.
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.
Partner Share! Example 1a
Simplify the expression.
There are no grouping symbols.
Rewrite division as multiplication.
Multiply.
48
Partner Share! Example 1b
Simplify the expression.
The square root sign acts as a
grouping symbol.
Subtract.
37
Take the square root.
21
Multiply.
Partner Share! Example 1c
Simplify the expression.
The division bar acts as a grouping
symbol.
Add and evaluate the power.
Multiply, subtract and simplify.
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.
Partner Share! 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.
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
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
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.
Additional Example 1: Identifying Properties
Name the property that is illustrated in each
equation.
A. 7(mn) = (7m)n
The grouping is different.
Associative Property of Multiplication
B. (a + 3) + b = a + (3 + b) The grouping is different.
Associative Property of Addition
C. x + (y + z) = x + (z + y) The order is different.
Commutative Property of Addition
Partner Share! Example 1
Name the property that is illustrated in each
equation.
a. n + (–7) = –7 + n
Commutative Property of Addition
b. 1.5 + (g + 2.3) = (1.5 + g) + 2.3
Associative Property of Addition
c. (xy)z = (yx)z
The order is
different.
The grouping is
different.
The order is
different.
Commutative Property of Multiplication
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.
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.
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.
Caution!
Add or subtract only the coefficients.
6.8y² – y² ≠ 6.8
Partner Share! 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.
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.
Partner Share! 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.