1-7 Simplifying Expressions

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California Standards

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1-7 Simplifying Expressions

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1. 42 2. |5 – 16|

3. –23

6. The difference of 10y and 4

5. The product of 8 and 6 8 6

–8

16 11

4. |3 – 7| 4

Translate each word phrase into a numerical or algebraic expression.

10y – 4

Simplify each fraction.

7. 8 8.

1-7 Simplifying Expressions

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.

California Standards

1-7 Simplifying Expressions

order of operationstermslike termscoefficient

Vocabulary

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

First:

Second:

Third:

Fourth:

Perform operations inside grouping symbols.

Perform multiplication and division from left to right.

Evaluate powers.

Perform addition and subtraction from left to right.

Order of Operations

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

A. 15 – 2 3 + 1

15 – 2 3 + 1 There are no grouping symbols.

15 – 6 + 1 Multiply.

9 + 1 Subtract.

10 Add.B. 12 + 32 + 10 ÷ 2

12 + 32 + 10 ÷ 2

12 + 9 + 10 ÷ 2

26

Additional Example 1: Simplifying Numerical Expressions

Simplify each expression.

There are no grouping symbols.Evaluate powers. The exponent

applies only to the 3.

Add.Divide.12 + 9 + 5

1-7 Simplifying Expressions

The fraction bar is a grouping symbol.

Multiply above the bar and subtract below the bar.

Add above the bar and then divide.

Evaluate powers. The exponent applies only to the 4.

C.

Additional Example 1: Simplifying Numerical Expressions

Simplify each expression.

1-7 Simplifying Expressions

Check It Out! Example 1a

Simplify the expression.

There are no grouping symbols.

Multiply.

Rewrite division as multiplication.

48

1-7 Simplifying Expressions

Check It Out! Example 1b

Simplify the expression.

The square root sign acts as a grouping symbol.

Subtract.

Take the square root.3 7

Multiply.21

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 ExpressionsAdditional 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)20 + 24 + 3579

A total of $79 was collected on Friday.

Substitute values for variables.Multiply.Add.

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)

223 + 46 + 14 + 117

400

Substitute values for variables.

Multiply.

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.

4x – 3x + 2

Like terms Constant

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.

1x2 + 3x

Coefficients

1-7 Simplifying Expressions

Like terms can be combined. To combine like terms, use the Distributive Property.

Notice that you can combine like terms by adding or subtracting the coefficients. Keep the variables and exponents the same.

= 3x

Distributive Property

ax – bx = (a – b)x

Example

7x – 4x = (7 – 4)x

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 ExpressionsAdditional Example 3: Combining Like Terms

Simplify the expression by combining like terms.

A variable without a coefficient has a coefficient of 1.

Write 1 as .

Add the coefficients.

and are like terms.

B.

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 + 2.5n

0.5m and 2.5n are not like terms.

Do not combine the terms.

1-7 Simplifying Expressions

Caution!Add or subtract only the coefficients.

6.8y² – y² ≠ 6.8

1-7 Simplifying ExpressionsCheck It Out! Example 3

Simplify by combining like terms.

a. 16p + 84p

16p + 84p

100p

16p + 84p are like terms.

Add the coefficients.

b. –20t – 8.5t

–20t – 8.5t 20t and 8.5t are like terms.

–28.5t Subtract the coefficients.

3m2 – m2 + m3 3m2 and – m2 are like terms.

c. 3m2 + m3 – m2

Subtract coefficients.2m2 + m3

1-7 Simplifying ExpressionsAdditional Example 4: Simplifying Algebraic Expressions

Use properties and operations to show that 14x + 4(2 + x) simplifies to 18x + 8.

14x + 4(2) + 4(x) Distributive Property

Multiply.

Commutative Property of AdditionAssociative Property of AdditionCombine like terms.

14x + 8 + 4x

(14x + 4x) + 8

14x + 4x + 8

18x + 8

14x + 4(2 + x)1. 2. 3. 4. 5. 6.

Statements Reasons

1-7 Simplifying Expressions

6x – 6(4) + 9 Distributive Property

Multiply.

Combine like terms.

6x – 24 + 9

6x – 15

6(x – 4) + 91. 2. 3. 4.

Statements Reasons

Check It Out! Example 4

Use properties and operations to show that 6(x – 4) + 9 simplifies to 6x – 15.

1-7 Simplifying ExpressionsLesson Quiz

Simplify each expression.

1. 165 + 27 + 3 + 5 2. 200 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.

6. Use properties and operations to show that 24a + b² + 3a + 2b² simplifies to 27a + 3b².

4. 5. 14c2 – 9c 14c2 – 9c

Check students’ work.