Algebraic Expressions 2x + 3y - 7
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Transcript of Algebraic Expressions 2x + 3y - 7
Algebraic Expressions
2x + 3y - 7What are the Terms?
Algebraic Expressions
2x + 3y - 7
Terms
Algebraic Expressions
2x + 3y - 7What are the variables?
Algebraic Expressions
2x + 3y - 7
Variables
Algebraic Expressions
2x + 3y - 7What are the coefficients?
Algebraic Expressions
2x + 3y - 7
Coefficients
Algebraic Expressions
2x + 3y - 7What is the constant?
Algebraic Expressions
2x + 3y - 7Constant
Algebraic ExpressionsPolynomial:monomial → x, 2xy, 4, 3x²y, … single termbinomial → x+1, 2xy+x, 3x²y+4, …two termstrinomial → 2x+3y+7, 3x²y+xy+4x, …three termspolynomial → …four or more terms
What is the area of a rectangle?
Length times WidthIf 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 - 33 • 2n4 1m
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
x3
Here are some phrases you may have see throughout the year. The terms with * are
ones that are often used.Addition Subtraction Multiplication Divisionsum* difference* product* quotient*increase decrease times dividedplus minus multiplied ratioadd subtractmore than less thantotal
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.
4n - 11
4) two more than 6 times a number.
6n + 2
5) the quotient of a number and 12.
12x
Which of the following expressions represents 7 times a number decreased by 13?
1. 7x + 132. 7x - 133. 13 - 7x4. 13 + 7x
Which one of the following expressions represents 28 less than three times a number?
1. 28 - 3x2. 3x - 283. 28 + 3x4. 3x + 28
Write a verbal expression for:
1) 8 + a.
The ratio of m to rDo you have a different way of writing these?
The sum of 8 and a2)
mr
.
Which of the following verbal expressions represents 2x + 9?
1. 9 increased by twice a number2. a number increased by nine3. twice a number decreased by 94. 9 less than twice a number
Which of the following expressions represents the sum of 16 and five times a number?
1. 5x - 162. 16x + 53. 16 + 5x4. 16 - 5x
Which of the following verbal expressions represents x2 + 2x?
1. the sum of a number squared and twice the number
2. the sum of a number and twice the number
3. twice a number less than the number squared
4. the sum of a number and twice the number squared
Which of the following expressions represents four less than the cube of a number?
1. 4 – x3
2. 4 – 3x3. 3x – 44. x3 – 4
Evaluate.21 222 2 • 2 = 423 2 • 2 • 2 = 82n7 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
Evaluate the following algebraic expression using
x=5, y=2
8(x-y)
Answer:
24
Evaluate the following algebraic expression using
x=7, y=2
yx ÷ 2
Answer:
7
Evaluate the following algebraic expression using
x=1, z=19
z + x³
Answer:
20
Evaluate the following algebraic expression using
m=3, p=10
15-(m+p)
Answer:
2
Evaluate the following algebraic expression using
a=9, b=4
b(a+b) + a
Answer:
61
Evaluate the following algebraic expression using
m=3, p=4
p²÷4-m
Answer:
1
Evaluate the following algebraic expression using
x=4, y=2
y(x-(9-4y))
Answer:
6
Evaluate the following algebraic expression using
x=9, y=1
x-(x-(x-y³))
Answer:
8
Evaluate the following algebraic expression using
h=9, j=8
j(h-9)³ +2
Answer:
2
Simplifying Algebraic Expressions
REVIEW
Vocabularytermcoefficientlike terms
Insert Lesson Title Here
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
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
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 –.
7x + 5 – 3y2 + y + x3
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
term
Term
Coefficient
4a 23 3k2 x2 x
9 4.7t
4 23 3 1 1
9 4.7
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
3x and 2x
5x2 and 2xThe exponentsare different.
3.2 and nOnly one term
contains avariable
6a and 6bThe variablesare different
w and w7 5 and 1.8
Identify like terms in the list.
Additional Example 1: Identifying Like Terms
3t 5w2 7t 9v 4w2 8v
Look for like variables with like powers.
3t 5w2 7t 9v 4w2 8v
Like terms: 3t and 7t, 5w2 and 4w2, 9v and 8v
Use different shapes or colors to indicate sets of like terms.Helpful Hint
Insert Lesson Title HereIdentify like terms in the list.
2x 4y3 8x 5z 5y3 8z
Look for like variables with like powers.
Like terms: 2x and 8x, 4y3 and 5y3 , 5z and 8z
2x 4y3 8x 5z 5y3 8z
Insert Lesson Title Here
x
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
4x + 5x = 9x
To combine like terms that have variables, add orsubtract the coefficients.
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
= (3)x2
= 3x2
Factor out x2 from both terms.
Perform operations in parenthesis.
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
47p
72p and 25p are like terms.
Subtract the coefficients.
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.
Simplify the expression by combining like terms.
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.
Simplify by combining like terms.
16p + 84p16p + 84p
100p
16p + 84p are like terms.
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 and m3 are not like terms.3m2 + m3
Do not combine the terms.3m2 + m3
Simplify 14x + 4(2 + x)
14x + 4(2) + 4(x) Distributive Property
Multiply.
Commutative Property
Associative Property
Combine like terms.
14x + 8 + 4x
(14x + 4x) + 8
14x + 4x + 8
18x + 8
14x + 4(2 + x)1.
2.
3.
4.
5.
6.
Procedure Justification
6(x) – 6(4) + 9 Distributive Property
Multiply.
Combine like terms.
6x – 24 + 9
6x – 15
6(x – 4) + 91.
2.
3.
4.
Procedure Justification
Simplify 6(x – 4) + 9. Justify each step.
–12x – 5x + x + 3a Commutative Property
Combine like terms.–16x + 3a
–12x – 5x + 3a + x1.
2.
3.
Procedure Justification
Simplify −12x – 5x + 3a + x. Justify each step.
Simplify each expression.
165 +27 + 3 + 5
Write each product using the Distributive Property. Then simplify.
5($1.99)
6(13)
200
8
5($2) – 5($0.01) = $9.95
6(10) + 6(3) = 78
Simplify each expression by combining like terms. Justify each step with an operation or property.
301x – x
24a + b2 + 3a + 2b2
300x
27a + 3b2
14c2 – 9c 14c2 – 9c
Let’s work more problems…
Simplify the following algebraic expression:
-3p + 6p
Answer:
3p
Simplify the following algebraic expression:
7x - x
Answer:
6x
Simplify the following algebraic expression:
-10v + 6v
Answer:
-4v
Simplify the following algebraic expression:
5n + 9n
Answer:
14n
Simplify the following algebraic expression:
b - 3 + 6 - 2b
Answer:
-b + 3
Simplify the following algebraic expression:
10x + 36 - 38x - 47
Answer:
-28x - 11
Simplify the following algebraic expression:
10x-w+4y-3x+36-38x-47+32x+2w-3y
Answer:
w+x+y-11
Simplify the following algebraic expression using the distributive property:
6(1 – 5m)
Answer:
6 – 30m
Simplify the following algebraic expression using the distributive property:
-2(1 – 5v)
Answer:
-2 + 10v
Simplify the following algebraic expression using the distributive property:
-3(7n + 1)
Answer:
-21n - 3
Simplify the following algebraic expression using the distributive property:
(x + 1) 14∙
Answer:
14x + 14
Simplify the following algebraic expression using the distributive property:
(3 - 7k) (-2)∙
Answer:
-6 + 14k
Simplify the following algebraic expression using the distributive property:
-20(8x + 20)
Answer:
-160x - 400
Simplify the following algebraic expression using the distributive property:
(7 + 19b) -15∙
Answer:
-105 – 285b
Variable Expressions
))()((
))()()()(()(
3
5
yyymeansy
xxxxxtionmultiplicaforsparentheseusemeansx
Simplify:
(-a)²
Answer:
a²
Substitution and EvaluatingSTEPS
1. Write out the original problem.2. Show the substitution with parentheses.3. Work out the problem.
3;4: xxifSolveExample 3;4 xxifSolve
3)4( = 64
Evaluate the variable expression when x = 1, y = 2, and w = -3
22 )()( yx
22 )()( yx
22 )2()1(
541
Step 1
Step 2
Step 3
2)( yx
2)( yx
2)2()1(
9)3( 2
Step 1
Step 2
Step 3
ywx
ywx
2)1)(3(
Step 1
Step 2
Step 3
3)1)(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 PROPERTIESPRODUCT OF POWERS
This property is used to combine 2 or more exponential expressions with the SAME base.
53 22 )222( )22222( 82 256
))(( 43 xx ))()(( xxx ))()()(( xxxx 7x
MULTIPLICATION PROPERTIESPOWER OF PRODUCT
This property combines the first 2 multiplication properties to simplify exponential expressions.
2)56( )5()6( 22 9002536
3)5( xy ))()(5( 333 yx33125 yx
532 )4( xx 5323 ))(4( xx 5222 ))()(()64( xxxx
56 ))(64( xx 1164x
Problems
Are you ready?3, 2, 1…lets go!
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
MULTIPLICATION PROPERTIESPOWER TO A POWER
This property is used to write and exponential expression as a single power of the base.
32 )5( )5)(5)(5( 222 65
42 )(x ))()()(( 2222 xxxx 8x
MULTIPLICATION PROPERTIESSUMMARY
PRODUCT OF POWERSbaba xxx
POWER TO A POWER baba xx
POWER OF PRODUCTaaa yxxy )(
ADD THE EXPONENTS
MULTIPLY THE EXPONENTS
Problems
Are you ready?3, 2, 1…lets go!
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 EXPONENTSANYTHING TO THE ZERO POWER IS 1.
271
313
91
313
31
313
13
33
93
273
33
22
11
0
1
2
3
22222
222
41
21
)2(1)2(
2122
xxxx
xxx
813131
311
311
31 4
4
4
4
4
DIVISION PROPERTIES QUOTIENT OF POWERS
This property is used when dividing two or more exponential expressions with the same base.
))()(())()()()((
3
5
xxxxxxxx
xx
2
1))(( xxx
7434
34
3
4
3 11111
xxxx
xxx
xx
DIVISION PROPERTIESPOWER OF A QUOTIENT
12
8
43
424
3
2
)()(
yx
yx
yx
Hard Example
3
43
2
32
yxxy
343
32
)3()2(
yxxy
1293
633
32
yxyx
69
123
278
yxyx
69
123
278
yxyx
6
6
278
xy
ZERO, NEGATIVE, AND DIVISION PROPERTIES
Zero power 1)( 0 x
Negative Exponents
aa
aa
xx
andx
x
1
1
Quotient of powers
bab
a
xxx
Power of a quotient
a
aa
yx
yx
Problems
Are you ready?3, 2, 1…lets go!
Simplify. Your answer should contain only positive exponents.
3r³2r
Answer:
3r²2
Simplify. Your answer should contain only positive exponents.
3xy 5x²( )
2
Answer:
9y²25x²
Simplify. Your answer should contain only positive exponents.
18x⁸y⁸ 10x³
Answer:
9x⁵y⁸5
Simplify:
(x⁴y¯²)(x¯¹y⁵)
Answer:
x³y³
Simplify the following algebraic expression using the distributive property:
8x (6x + 6)∙
Answer:
48x² + 48x
Simplify the following algebraic expression using the distributive property:
7n(6n + 3)
Answer:
42n² + 21n
Simplify the following algebraic expression using the distributive property:
2(9x – 2y)
Answer:
18x – 4y
Simplify the following algebraic expression using the distributive property:
1 + 7(1 – 3b)
Answer:
8 - 21b
Simplify the following algebraic expression using the distributive property:
-2 - 7(-1 – 3b)
Answer:
5 + 21b
Simplify the following algebraic expression using the distributive property:
3n(n² - 6n + 5)
Answer:
3n³ - 18n² + 15n
Simplify the following algebraic expression using the distributive property:
2k³(2k² + 5k - 4)
Answer:
4k⁵ +10k⁴ - 8k³
Simplify the following algebraic expression using the distributive property:
9(x² + xy – 8y²)
Answer:
9x² + 9xy – 72y²
Simplify the following algebraic expression using the distributive property:
9v²(u² + uv - 5v²)
Answer:
9v²u² +9v³u – 45v⁴
Simplify the following algebraic expression using the distributive property:
3x(5x+2) - 14(2x²-x+1)
Answer:
-13x² + 20x - 14
Simplify completely:
4x²y2x
Answer:
2xy
Simplify completely:
y¯¹ y¯²
Answer:
y
Simplify completely:
16x⁴y¯¹4x²y¯²
Answer:
4x²y
Simplify completely:
36x³y⁶z¹²4x¯¹y³z¹⁰
Answer:
9x⁴y³z²
Simplify completely:
21x³y⁷z¹⁴ 30x³z¯⁵18x⁴y⁶ y¹²z¯⁶·
Answer:
35x²z¹⁵y¹¹