Contributed by:

This pdf includes the following topics:-

Expressions

Variables

Algebraic Expression

Expressions vs. Equations

Coefficients

Terms

Expressions

Variables

Algebraic Expression

Expressions vs. Equations

Coefficients

Terms

1.
Variables and Expressions,

Order of Operations

Notes 1.1Expressions

and 1.2 and Terms

Order of Operations

Notes 1.1Expressions

and 1.2 and Terms

2.
I. Expressions

You are familiar with the following type of

numerical expressions:

12 + 6

3 (12)

6 (3 + 2)

15 - 4 (6)

You are familiar with the following type of

numerical expressions:

12 + 6

3 (12)

6 (3 + 2)

15 - 4 (6)

3.
II. Variables

In the expression 12 + B,

the letter “B” is a variable.

Definition:

A variable is a letter or symbol that

represents an unknown value.

In the expression 12 + B,

the letter “B” is a variable.

Definition:

A variable is a letter or symbol that

represents an unknown value.

4.
III. Algebraic Expressions

When variables are used with other numbers,

parentheses, or operations, they create an

algebraic expression.

a + 2

(a) (b)

3m + 6n - 6

When variables are used with other numbers,

parentheses, or operations, they create an

algebraic expression.

a + 2

(a) (b)

3m + 6n - 6

5.
Expressions vs. Equations

Expressions Equations

No = sign = sign

Simplify Solve

Expressions Equations

No = sign = sign

Simplify Solve

6.
IV. Coefficients

A coefficient is the number multiplied by the variable

in an algebraic expression.

Algebraic Coefficients

6m + 5 6

8r + 7m + 4 8, 7

14b - 8 14

A coefficient is the number multiplied by the variable

in an algebraic expression.

Algebraic Coefficients

6m + 5 6

8r + 7m + 4 8, 7

14b - 8 14

7.
IV. Terms

A term is the name given to a number, a

variable, or a number and a variable combined

by multiplication or division.

Algebraic Terms

a + 2 a, 2

3m + 6n - 6 3m, 6n, - 6

A term is the name given to a number, a

variable, or a number and a variable combined

by multiplication or division.

Algebraic Terms

a + 2 a, 2

3m + 6n - 6 3m, 6n, - 6

8.
V. Constants

• A constant is a number that cannot change its

value.

In the expression: 5x + 7y + 2

the constant is 2

In the expression: x - 3

the constant is -3

• A constant is a number that cannot change its

value.

In the expression: 5x + 7y + 2

the constant is 2

In the expression: x - 3

the constant is -3

9.
VI. Factors and Products

(3)(5) 15

?

These are factors This is a product

Factors: Quantities being multiplied

Products: The result of multiplying factors

(3)(5) 15

?

These are factors This is a product

Factors: Quantities being multiplied

Products: The result of multiplying factors

10.
VII. Powers, Bases and Exponents

2 7

4 6 This is an

exponent

This is a Power This is a base

2 7

4 6 This is an

exponent

This is a Power This is a base

11.
Let’s Practice!

Identify the terms, coefficients, and constants.

TERMS COEFFICIENTS CONSTANTS

12a – 6b + 4 12a, -6b, 4 12,-6 4

4x – 2y 4x, -2y 4, -2 0

C - 32

C, -32 1 -32

3x + 2 3x, 2 3 2

Identify the terms, coefficients, and constants.

TERMS COEFFICIENTS CONSTANTS

12a – 6b + 4 12a, -6b, 4 12,-6 4

4x – 2y 4x, -2y 4, -2 0

C - 32

C, -32 1 -32

3x + 2 3x, 2 3 2

12.
VIII. Writing Algebraic Expressions

• A. You can translate word phrases into variable

expressions.

– Examples:

1. Three more than a number = x + 3

2. The quotient of a number and 8 = y/8

3. Six times a number = 6 • n or 6n

4. 15 less than a number = z – 15

5. The quotient of 30 and a number plus 10 = 30/x

+ 10.

• A. You can translate word phrases into variable

expressions.

– Examples:

1. Three more than a number = x + 3

2. The quotient of a number and 8 = y/8

3. Six times a number = 6 • n or 6n

4. 15 less than a number = z – 15

5. The quotient of 30 and a number plus 10 = 30/x

+ 10.

13.
B. Key words to look for:

• Addition: • Subtraction:

– Add – Minus

– Plus – Difference

– Sum – Subtract

– Total – Less than

– Increased by – Decreased by

– More than – less

• Addition: • Subtraction:

– Add – Minus

– Plus – Difference

– Sum – Subtract

– Total – Less than

– Increased by – Decreased by

– More than – less

14.
Cont…

• Multiplication • Division

– Product – Quotient

– Times – Divide

– Multiply – Split Between

– per

• Multiplication • Division

– Product – Quotient

– Times – Divide

– Multiply – Split Between

– per

15.
C. Write algebraic expressions for these

word phrases

1. Four more than s

2. The product of 7 and c

3. Nine less than x

4. A number divided by the sum of 4 and 7.

5. Twice the sum of a number plus 4.

6. The sum of ¾ of a number and 7.

7. Ten times a number increased by 150.

word phrases

1. Four more than s

2. The product of 7 and c

3. Nine less than x

4. A number divided by the sum of 4 and 7.

5. Twice the sum of a number plus 4.

6. The sum of ¾ of a number and 7.

7. Ten times a number increased by 150.

16.
D. Write an algebraic phrase for these

situations

1. A car was traveling 35 miles per hour

for a number of hours.

2. Bob ran 7 times a week for a number of

weeks.

3. The plumber added an extra $35 to her

bill.

4. Thirty-five fewer people came than the

number expected.

situations

1. A car was traveling 35 miles per hour

for a number of hours.

2. Bob ran 7 times a week for a number of

weeks.

3. The plumber added an extra $35 to her

bill.

4. Thirty-five fewer people came than the

number expected.

17.
IX. ORDER OF OPERATIONS

When an expression contains more than one

operation, the order of operations tells you

which operation to perform first.

When an expression contains more than one

operation, the order of operations tells you

which operation to perform first.

18.
Order of Operations

1. Grouping: Perform operations inside grouping symbols.

2. Powers: Evaluate all powers.

3. Mult/Div: Multiply and/or divide in order from left to right.

4. Add/Sub: Add and/or subtract from left to right.

1. Grouping: Perform operations inside grouping symbols.

2. Powers: Evaluate all powers.

3. Mult/Div: Multiply and/or divide in order from left to right.

4. Add/Sub: Add and/or subtract from left to right.

19.
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.

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.

20.
Ex 1: Simplifying Numerical Expressions

Simplify each expression.

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 There are no grouping symbols.

12 + 9 + 10 ÷ 2 Evaluate powers. The exponent

applies only to the 3.

12 + 9 + 5 Divide.

26 Add.

Simplify each expression.

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 There are no grouping symbols.

12 + 9 + 10 ÷ 2 Evaluate powers. The exponent

applies only to the 3.

12 + 9 + 5 Divide.

26 Add.

21.
Simplify each expression.

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.

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.

22.
Simplify the expression.

There are no grouping symbols.

Rewrite division as multiplication.

Multiply.

48

There are no grouping symbols.

Rewrite division as multiplication.

Multiply.

48

23.
E. Simplify the expression.

The square root sign acts as a

grouping symbol.

Subtract.

37 Take the square root.

21 Multiply.

The square root sign acts as a

grouping symbol.

Subtract.

37 Take the square root.

21 Multiply.

24.
F. Simplify the expression.

The division bar acts as a grouping

symbol.

Add and evaluate the power.

Multiply, subtract and simplify.

The division bar acts as a grouping

symbol.

Add and evaluate the power.

Multiply, subtract and simplify.

25.
X. Evaluating Algebraic Expressions

Evaluate a2 – (b3-4c) if a = 7, b = 3, and c = 5.

• To evaluate an algebraic expression, replace

the variables with their values. Then, find

the value of the numerical expression using

order of operations.

Evaluate a2 – (b3-4c) if a = 7, b = 3, and c = 5.

• To evaluate an algebraic expression, replace

the variables with their values. Then, find

the value of the numerical expression using

order of operations.

26.
A. Evaluate a2 – (b3 - 4c) if a = 7, b = 3, and c = 5.

a2 – (b3 - 4c) = 72 – (33 – 4 • 5) Replace a with 7, b

with 3, and c with 5.

= 49 – (27 – 4 • 5) Evaluate 72 and 33

Multiply 4 and 5

= 49 – (27 – 20)

Subtract 20 from 27

= 49 – 7

Subtract

= 42

a2 – (b3 - 4c) = 72 – (33 – 4 • 5) Replace a with 7, b

with 3, and c with 5.

= 49 – (27 – 4 • 5) Evaluate 72 and 33

Multiply 4 and 5

= 49 – (27 – 20)

Subtract 20 from 27

= 49 – 7

Subtract

= 42

27.
B. • Evaluate x(y3 + 8) ÷ 12 if x = 3, and y = 4.

x(y3 + 8) ÷ 12 = 3(43 + 8) ÷ 12 Replace x with 3 and

y with 4.

= 3(64 + 8) ÷ 12 Evaluate 43

= 3(72) ÷ 12 Add 64 and 8

216 ÷ 12 Multiply 72 and 3

= 18 Divide

x(y3 + 8) ÷ 12 = 3(43 + 8) ÷ 12 Replace x with 3 and

y with 4.

= 3(64 + 8) ÷ 12 Evaluate 43

= 3(72) ÷ 12 Add 64 and 8

216 ÷ 12 Multiply 72 and 3

= 18 Divide

28.
Business Connection: Consumerism

C. According to market research, the average consumer spends

$78 per trip to the mall on the weekends and only $67 per

trip during the week.

--Write an algebraic expression to represent how much the average

consumer spends at the mall in x weekend trips and y weekday trips.

78x + 67y

--Evaluate the expression to find what the average consumer spends

after going to the mall twice during the week and 5 times on the

weekends.

The average consumer

78(5) + 67(2)= spends $554.

390 + 134= 554

C. According to market research, the average consumer spends

$78 per trip to the mall on the weekends and only $67 per

trip during the week.

--Write an algebraic expression to represent how much the average

consumer spends at the mall in x weekend trips and y weekday trips.

78x + 67y

--Evaluate the expression to find what the average consumer spends

after going to the mall twice during the week and 5 times on the

weekends.

The average consumer

78(5) + 67(2)= spends $554.

390 + 134= 554