Algebraic Expressions 2x + 3y - 7

Algebraic Expressions 2x + 3y - 7
What are the Terms?

Terms

What are the variables?

Variables

What are the coefficients?

Coefficients

What is the constant?

Constant

Algebraic Expressions Polynomial: monomial → x, 2xy, 4, 3x²y, … single term binomial → x+1, 2xy+x, 3x²y+4, …two terms trinomial → 2x+3y+7, 3x²y+xy+4x, …three terms polynomial → …four or more terms

What is the area of a rectangle?
Length times Width If the length is 3 meters and the width is 2 meters, what is the area? A = L x W A = 3 x 2 = 6 meters2 A, L and W are the variables. It is any letter that represents an unknown number.

An algebraic expression contains:
1) one or more numbers or variables, and 2) one or more arithmetic operations. Examples: x - 3 3 • 2n

In expressions, there are many different ways to write multiplication.
1) ab 2) a • b 3) a(b) or (a)b 4) (a)(b) 5) a x b We are not going to use the multiplication symbol any more. Why?

Division, on the other hand, is written as:
1) 2) x ÷ 3

Here are some phrases you may have see throughout the year
Here are some phrases you may have see throughout the year. The terms with * are ones that are often used.

Write an algebraic expression for 1) m increased by 5.
m + 5 2) 7 times the product of x and t. 7xt or 7(x)(t) or 7 • x • t

3) 11 less than 4 times a number.
4) two more than 6 times a number. 6n + 2 5) the quotient of a number and 12.

Which of the following expressions represents 7 times a number decreased by 13?

Which one of the following expressions represents 28 less than three times a number?

Write a verbal expression for: 1) 8 + a.
The sum of 8 and a 2) The ratio of m to r Do you have a different way of writing these?

Which of the following verbal expressions represents 2x + 9?
9 increased by twice a number a number increased by nine twice a number decreased by 9 9 less than twice a number

Which of the following expressions represents the sum of 16 and five times a number?

Which of the following verbal expressions represents x2 + 2x?
the sum of a number squared and twice the number the sum of a number and twice the number twice a number less than the number squared the sum of a number and twice the number squared

Which of the following expressions represents four less than the cube of a number?

Evaluate. 21 • 2 = • 2 • 2 = 8 2n7 We can’t evaluate because we don’t know what n equals to!!

Competition Problems Evaluating Algebraic Expressions

Evaluate the following algebraic expression using
m=7, n=8 n² - m

Answer: 57

x=5, y=2 8(x-y)

Answer: 24

x=7, y=2 yx ÷ 2

Answer: 7

x=1, z=19 z + x³

Answer: 20

m=3, p=10 15-(m+p)

Answer: 2

a=9, b=4 b(a+b) + a

Answer: 61

m=3, p=4 p²÷4-m

Answer: 1

x=4, y=2 y(x-(9-4y))

Answer: 6

x=9, y=1 x-(x-(x-y³))

Answer: 8

h=9, j=8 j(h-9)³ +2

Answer: 2

Simplifying Algebraic Expressions

REVIEW

Insert Lesson Title Here
Vocabulary term coefficient like terms

The terms of an expression are the parts to be added or subtracted
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
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

In the expression 7x + 5, 7x and 5 are called terms
In the expression 7x + 5, 7x and 5 are called terms. A term can be a number, a variable, or a product of numbers and variables. Terms in an expression are separated by + and –. x 3 7x – y y + term term term term term In the term 7x, 7 is called the coefficient. A coefficient is a number that is multiplied by a variable in an algebraic expression. A variable by itself, like y, has a coefficient of 1. So y = 1y. Coefficient Variable 7 x

Term Coefficient 2 3 x 9 4a 3k2 x2 4.7t 2 3 1 9 4 3 1 4.7

Like terms are terms with the same variable raised to the same power
Like terms are terms with the same variable raised to the same power. The coefficients do not have to be the same. Constants, like 5, , and 3.2, are also like terms. 1 2 Like Terms Unlike Terms w 7 3x and 2x w and 5 and 1.8 5x2 and 2x 6a and 6b 3.2 and n Only one term contains a variable The exponents are different. The variables are different

Additional Example 1: Identifying Like Terms
Identify like terms in the list. 3t 5w2 7t 9v w2 8v Look for like variables with like powers. 3t w t v w v Like terms: 3t and 7t, 5w2 and 4w2, 9v and 8v Use different shapes or colors to indicate sets of like terms. Helpful Hint

Identify like terms in the list.
Insert Lesson Title Here Identify like terms in the list. 2x 4y3 8x 5z y3 8z Look for like variables with like powers. 2x y x z y z Like terms: 2x and 8x, 4y3 and 5y3 , 5z and 8z

Insert Lesson Title Here
Combining like terms is like grouping similar objects. x x x x x x x x + = x x x x x x x x x x = 9x 4x + 5x To combine like terms that have variables, add or subtract the coefficients.

Using the Distributive Property can help you combine like terms
Using the Distributive Property can help you combine like terms. You can factor out the common factors to simplify the expression. 7x2 – 4x2 = (7 – 4)x2 Factor out x2 from both terms. = (3)x2 Perform operations in parenthesis. = 3x2 Notice that you can combine like terms by adding or subtracting the coefficients and keeping the variables and exponents the same.

Simplify the expression by combining like terms.
72p – 25p 72p – 25p 72p and 25p are like terms. 47p Subtract the coefficients.

A variable without a coefficient has a coefficient of 1. and are like terms. Write 1 as Add the coefficients.

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.

16p + 84p –20t – 8.5t2 3m2 + m3 Simplify by combining like terms.
16p + 84p are like terms. 100p Add the coefficients. –20t – 8.5t2 –20t – 8.5t2 20t and 8.5t2 are not like terms. –20t – 8.5t2 Do not combine the terms. 3m2 + m3 3m2 + m3 3m2 and m3 are not like terms. 3m2 + m3 Do not combine the terms.

Simplify 14x + 4(2 + x) Procedure Justification 1. 14x + 4(2 + x) 2.
Distributive Property 3. 14x x Multiply. Commutative Property 4. 14x + 4x + 8 5. (14x + 4x) + 8 Associative Property 6. 18x + 8 Combine like terms.

Simplify 6(x – 4) + 9. Justify each step.
Procedure Justification 1. 6(x – 4) + 9 2. 6(x) – 6(4) + 9 Distributive Property 3. 6x – Multiply. Combine like terms. 4. 6x – 15

Simplify −12x – 5x + 3a + x. Justify each step.
Procedure Justification 1. –12x – 5x + 3a + x 2. –12x – 5x + x + 3a Commutative Property 3. –16x + 3a Combine like terms.

5($1.99) 6(13) 165 +27 + 3 + 5 Simplify each expression. 200 8
Write each product using the Distributive Property. Then simplify. 5($1.99) 5($2) – 5($0.01) = $9.95 6(13) 6(10) + 6(3) = 78

Simplify each expression by combining like terms
Simplify each expression by combining like terms. Justify each step with an operation or property. 14c2 – 9c 14c2 – 9c 301x – x 300x 24a + b2 + 3a + 2b2 27a + 3b2

Let’s work more problems…

Simplify the following algebraic expression:
-3p + 6p

Answer: 3p

7x - x

Answer: 6x

-10v + 6v

Answer: -4v

5n + 9n

Answer: 14n

b b

Answer: -b + 3

10x x - 47

Answer: -28x - 11

10x-w+4y-3x+36-38x-47+32x+2w-3y

Answer: w+x+y-11

Simplify the following algebraic expression using the distributive property:

Answer: 6 – 30m

Answer: v

Answer: -21n - 3

Answer: 14x + 14

(3 - 7k) ∙ (-2)

Answer: k

Answer: -160x - 400

Answer: – 285b

Variable Expressions

Simplify: (-a)²

Answer: a²

Substitution and Evaluating
STEPS Write out the original problem. Show the substitution with parentheses. Work out the problem. = 64

Evaluate the variable expression when x = 1, y = 2, and w = -3
Step 1 Step 1 Step 1 Step 2 Step 2 Step 2 Step 3 Step 3 Step 3

Contest Problem

Are you ready? 3, 2, 1…lets go!

Evaluate the expression when a= -2 a² + 2a - 6

Answer: -6

Evaluate the expression when x= -4 and t=2 x²(x-t)

Answer: -96

Evaluate the expression when y= -3 (2y + 5)²

Answer: 1

MULTIPLICATION PROPERTIES
PRODUCT OF POWERS This property is used to combine 2 or more exponential expressions with the SAME base.

POWER OF PRODUCT This property combines the first 2 multiplication properties to simplify exponential expressions.

Problems Are you ready? 3, 2, 1…lets go!

Simplify. Your answer should contain only positive exponents
Simplify. Your answer should contain only positive exponents. 2n⁴ · 5n ⁴

Answer: 10n⁸

Simplify. Your answer should contain only positive exponents. 6r · 5r²

Answer: 30r³

Simplify. Your answer should contain only positive exponents. 6x · 2x²

Answer: 12x³

Simplify. Your answer should contain only positive exponents. 6x² · 6x³y⁴

Answer: 36x⁵y⁴

Simplify. Your answer should contain only positive exponents. 10xy³ · 8x⁵y³

Answer: 80x⁶y⁶

POWER TO A POWER This property is used to write and exponential expression as a single power of the base.

SUMMARY PRODUCT OF POWERS ADD THE EXPONENTS POWER TO A POWER MULTIPLY THE EXPONENTS POWER OF PRODUCT

Simplify. Your answer should contain only positive exponents. (a²)³

Answer: a⁶

Simplify. Your answer should contain only positive exponents. (3a²)³

Answer: 27a⁶

Simplify. Your answer should contain only positive exponents. (x⁴y⁴)³

Answer: x¹²y¹²

Simplify. Your answer should contain only positive exponents. (2x⁴y⁴)³

Answer: 8x¹²y¹²

Simplify. Your answer should contain only positive exponents. (4x⁴∙x⁴)³

Answer: 64x²⁴

Simplify. Your answer should contain only positive exponents. (4n⁴∙n)²

Answer: 16n¹⁰

ZERO AND NEGATIVE EXPONENTS
ANYTHING TO THE ZERO POWER IS 1.

DIVISION PROPERTIES QUOTIENT OF POWERS
This property is used when dividing two or more exponential expressions with the same base.

DIVISION PROPERTIES POWER OF A QUOTIENT Hard Example

ZERO, NEGATIVE, AND DIVISION PROPERTIES
Zero power Quotient of powers Negative Exponents Power of a quotient

Simplify. Your answer should contain only positive exponents. 3r³ 2r

Answer: 3r² 2

Simplify. Your answer should contain only positive exponents. 3xy 5x²
2 ( )

Answer: 9y² 25x²

Simplify. Your answer should contain only positive exponents. 18x⁸y⁸ 10x³

Answer: 9x⁵y⁸ 5

Simplify: (x⁴y¯²)(x¯¹y⁵)

Answer: x³y³

8x ∙ (6x + 6)

Answer: 48x² + 48x

7n(6n + 3)

Answer: 42n² + 21n

2(9x – 2y)

Answer: 18x – 4y

Answer: b

3n(n² - 6n + 5)

Answer: 3n³ - 18n² + 15n

2k³(2k² + 5k - 4)

Answer: 4k⁵ +10k⁴ - 8k³

9(x² + xy – 8y²)

Answer: 9x² + 9xy – 72y²

9v²(u² + uv - 5v²)

Answer: 9v²u² +9v³u – 45v⁴

3x(5x+2) - 14(2x²-x+1)

Answer: -13x² + 20x - 14

Simplify completely: 4x²y 2x

Answer: 2xy

Simplify completely: y¯¹ y¯²

Answer: y

Simplify completely: 16x⁴y¯¹ 4x²y¯²

Answer: 4x²y

Simplify completely: 36x³y⁶z¹² 4x¯¹y³z¹⁰

Answer: 9x⁴y³z²

Simplify completely: 21x³y⁷z¹⁴ x³z¯⁵ 18x⁴y⁶ y¹²z¯⁶

Answer: 35x²z¹⁵ y¹¹

Algebraic Expressions 2x + 3y - 7

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