1
8th Grade Math
Solving Equations
2015­12­17
www.njctl.org
2
Table of Contents
Click on a topic to go to that section.
Multi­Step Equations
Solving Equations that Contain Fractions
Equations with the Same Variable on Both Sides
Comparing Expressions with the Same Variable
Writing and Solving Algebraic Equations
Teacher Notes
Review of Two Step Equations
Vocabu
in the p
box the
linked of the p
word d
Translating & Solving Consecutive Integer Problems
Glossary & Standards
3
Review of Two­Step Equations
Return to Table of Contents
4
Two­Step Equations
A two­step equation is an equation that contains two operations. For example, it could contain multiplication and subtraction, like the equation below.
5x ­ 9 = 16
Or it could contain addition and division like this equation.
x + 11 = ­6
4
Before we start solving two­step equations, let's review some tips for solving them.
5
Tips for Solving Equations
1. To "undo" a mathematical operation, you must perform the inverse operation. 2. You can do anything you want (except divide by zero) to one side of an equation, as long as you do the same thing to the other.
3. If there is more than one operation going on, you must undo them in the opposite order in which you would do them, the opposite of the "order of operations."
4. You can always switch the left and right sides of an equation.
6
Tips Explained
1. To "undo" a mathematical operation, you must do the opposite. We learned earlier that for every mathematics operation, there is an inverse operation which undoes it: when you do both operations, you get back to where you started.
When the variable for which we are solving is connected to something else by a mathematical operation, we can eliminate that connection by using the inverse of that operation.
7
Tips Explained
2. You can do anything you want (except divide by zero) to one side of an equation, as long as you do the same to the other side.
If the two expressions on the opposite sides of the equal sign are equal to begin with, they will continue to be equal if you do the same mathematical operation to both of them.
This allows you to use an inverse operation on one side, to undo an operation, as long as you also do it on the other side.
You can just never divide by zero (or by something which turns out to be zero) since the result of that is always undefined.
8
Tips Explained
3. If there is more than one operation going on, you must undo them in the opposite order in which you would do them, the opposite of the "order of operations."
The operations which are connected to a variable must be "undone" in the reverse order from the Order of Operations.
So, when solving for a variable, you:
first have to undo addition/subtraction, then multiplication/division,
then exponents/roots, finally parentheses. The order of the steps you take to untie a knot are the reverse of the order used to tie it.
9
Tips Explained
4. You can always switch the left and right sides of an equation.
Once an equation has been solved for a variable, it is typically easier to use if that variable is moved to the left side.
Mathematically, this has no effect since the both sides are equal.
10
Solving for y
Let's solve this equation for "y"
­4y ­ 11 = ­27
That means that when we're done
we'll have y alone
on the left side of the equation. 11
1 Is y already alone? If not, what is with it? Select all that apply.
A ­4
C ­11
Answer
B y
D ­27
E it is already alone
­4y ­ 11 = ­27
12
2 Which math operations connect the numbers to y? Select all that apply.
B Subtraction
C Multiplication
Answer
A Addition
D Division
­4y ­ 11 = ­27
13
A
B
C
D
Addition
Subtraction
Multiplication
Division
Answer
3 Which math operation gets undone first?
­4y ­ 11 = ­27
14
4 What must we do if we add 11 to the left side?
A Subtract 11 from the left side
C Add 11 to the left side
Answer
B Subtract 11 from the right side
D Add 11 to the right side
­4y ­ 11 = ­27
15
Solving for y
1. To "undo" a mathematical operation, you must do the opposite.
2. You can do anything you want (except divide by zero) to one side of an equation, as long as you do the same thing to the other.
So we undo 9 being subtracted from ­2y by adding 9 to both sides.
­4y ­ 11 = ­27
+ 11 +11
­4y = ­16
Answer
­4y ­ 11 = ­27
Are we done?
16
5 What math operation connects ­4 and y?
A ­4 is being added to y
C ­4 is being multiplied by y
D ­4 is being divided by y
Answer
B ­4 is being subtracted by y
­4y = ­16
17
6 What is the opposite of multiplying y by ­4?
A Dividing y by 4
C Multiplying y by 4 Answer
B Dividing y by ­4
D Multiplying y by ­4
­4y = ­16
18
Solving for y
1. To "undo" a mathematical operation, you must do the opposite.
­4y = ­16
2. You can do anything you want (except divide by zero) to one side of an equation, as long as you do the same thing to the other.
­4y = ­16
­4 ­4
y = 4
19
A
B
C
D
­11
­4
y
it is alone
Answer
7 Is y alone on the left? If not, what is with it? y = 4
20
Solving for t
Let's solve this equation for "t"
5 = 15
t
That means that when we're done
we'll have t alone
on the left side of the equation. 21
A 5
B 15
C t
D it is already alone
5 =
Answer
8 Is t already alone? If not, what is with it?
15
t
22
A
B
C
D
t is being divided by 15
15 is being divided by t
15 is being multiplied by t
t is being subtracted from 15
5 =
Answer
9 What mathematical operation connects d to t?
15
t
23
A dividing 15 by t
B dividing 5 by t
C multiplying 15 by t
D multiplying t by 15
5 =
15
t
Answer
10 What is the opposite of dividing 15 by t?
Rule 1. To "undo" a mathematical operation, you must do the opposite.
24
A divide the left side by t
B multiply the left side by t
C divide the left side by 15
D divide the left side by 5
5 =
Answer
11 What must we do if we multiply the right side by t?
15
t
Rule 2. You can do anything you want (except divide by zero) to one side of an equation, as long as you do the same thing to the other.
25
12 Is there more than one mathematical operation acting on "t"?
5 =
15
t
Answer
Yes No Rule 3. If there is more than one operation going on, you must undo them in the opposite order in which you would do them, the opposite of the "order of operations."
26
Solving for t
1. To "undo" a mathematical operation, you must do the opposite.
5 = 15
2. You can do anything you want (except divide by zero) to one side of an equation, as long as you do the same thing to the other.
So we undo d being divided by t, by multiplying both sides by t.
(t) 5 = 15
Answer
t
(t)
t
5t = 15
Are we done?
27
A
B
C
D
t is being divided by 15
t is being divided into 5
t is being multiplied by 5
t is being subtracted from 5
Answer
13 What mathematical operation connects 5 to t?
5t = 15
28
A dividing t by 5
B dividing t by t
C multiplying t by t
D multiplying t by 5
Answer
14 What is the opposite of multiplying t by 5?
5t = 15
29
Solving for t
1. To "undo" a mathematical operation, you must do the opposite.
5t = 15
5
2. You can do anything you want (except divide by zero) to one side of an equation, as long as you do the same thing to the other.
5t 15
=
5
5
t = 3
30
15 Is t alone on the left? If not, what is with it? 5
15
t
it is alone
Answer
A
B
C
D
t = 3
31
Review of Solving Two­Step Equations
The following formative assessment questions are review from 7th grade. If further instruction is need, see the presentation at:
https://www.njctl.org/courses/math/7th­grade/
equations­inequalities­7th­grade/
32
16 Solve the equation.
Answer
5x ­ 6 = ­56
33
17 Solve the equation.
Answer
14 = 3c + 2
34
18 Solve the equation.
Answer
x ­ 4 = 24
5
35
19 Solve the equation.
Answer
5r ­ 2 = ­12
36
20 Solve the equation.
Answer
14 = ­2n ­ 6
37
21 Solve the equation.
Answer
x + 7 = 13
5
38
22 Solve the equation.
Answer
x ­ + 2 = ­10
3
39
Answer
23 Solve the equation.
­2.5x ­ 4 = 3.5
40
Answer
24 Solve the equation.
3.3x ­ 4 = ­13.9
41
Answer
25 Solve the equation.
­x + (­5.1) = ­2.3
6
42
Answer
26 Solve the equation.
2.8x ­ 7 = ­1.4
43
Multi­Step Equations
Return to Table of Contents
44
Steps for Solving Multiple Step Equations
The a
Ask: Ho
1. Simplify each side of the equation.
(Combine like terms and use the distributive property.) 2. Use inverse operations to solve the equation.
Remember, whatever you do to one side of an equation, you MUST do to the other side!
Math Practice
As equations become more complex, you should:
Wha
Which i
Wha
operat
Emp
operatio
45
Multiple Step Equations
Example:
Answer
12h ­ 10h + 7 = 25
46
Multiple Step Equations
Example:
Answer
17 ­ 9f + 6 = 140
47
Hint
Always check to see that both sides of the equation are simplified before you begin solving the equation.
When an equation is simplified there should be at most one term for each variable and one constant term.
48
Distributive Property
In order to solve for a variable, it can't be in parentheses with other variables or numbers. If that is how the equation is given, you have to use the distributive property to get it on its own.
For example: You need to use the distributive property as shown below as a first step in solving for x.
5(x ­ 6) ­ 3x = 8
Distributive Property
5x ­ 30 ­ 3x = 8
Now we can combine like terms and solve
49
Distributive Property
Remember: The distributive property is a(b + c) = ab + ac
Examples:
5(20 + 6) = 5(20) + 5(6) 9(30 ­ 2) = 9(30) ­ 9(2) 3(5 + 2x) = 3(5) + 3(2x)
­2(4x ­ 7) = ­2(4x) ­ (­2)(7)
50
Distributive Property
Example:
Answer
3(w ­ 2) = 9
51
Distributive Property
Example:
Answer
6m + 2(2m + 7) = 54
52
27 Solve. Answer
3 + 2t + 4t = ­63
53
A
­0.4
B
5
C
­4
D
4
Answer
28 What is the value of n in the equation 0.6(n + 10) = 3.6? From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011
54
29 Solve.
Answer
19 = 1 + 4 ­ x
55
30 Solve.
Answer
8x ­ 4 ­ 2x ­ 11 = ­27
56
31 Solve.
Answer
­4 = ­27y + 7 ­ (­15y) + 13
57
32 Solve.
Answer
9 ­ 4y + 16 + 11y = 4
58
33 Solve.
Answer
6(­8 + 3b) = 78
59
34 Solve.
Answer
18 = ­6(1 ­ 1k) 60
35 Solve. Answer
2w + 8(w + 3) = 34
61
36 Solve.
Answer
4 = 4x ­ 2(x + 6)
62
37 Solve. Answer
3r ­ r + 2(r + 4) = 24
63
A
1
B
2 1/3
C
3
D
1/3
Answer
38 What is the value of p in the equation 2(3p ­ 4) = 10? From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
64
Solving Equations that Contain Fractions
Return to Table of Contents
65
Distributing Fractions
The a
Ask: Ho
Also, it's often easier to just get rid of fractions as a first step.
Math Practice
You always need to have the variable for which you are solving end up in the numerator.
Wha
Which i
Wha
operat
Emp
operatio
66
Distributing Fractions
3 (­3 + 3x) = 72 5
5
Multiply both sides of this equation by the least common denominator (LCD) of both sides to eliminate the fractions. Then this just becomes like all the equations we've been solving.
In this case, the LCD is 5...so just multiply both sides by 5 to get:
3(­3 + 3x) = 72
Distribute
­9 + 9x = 72
Add 9 to both sides
9x = 81
Divide both sides by 9
x = 9
67
Distributing Fractions
Example:
4 (x + 9) = 2 7
3
Multiply both sides by 21
(21)(4) (x + 9) = (2)(21)
3
7
Simplify fractions
12(x + 9) = 14
Distribute the 12
12x + 108 = 14
Subtract 108 from both sides
12x = ­94
Divide both sides by 12
x = ­7 10/12
Simplify fraction
x = ­7 5/6
68
Distributing Fractions
Answer
Example:
3 (x + 2) = 9
5
69
Distributing Fractions
Example:
Answer
1 (x + ) =
1 7 8
2
16
70
39 Solve the equation. Answer
2x ­ 3 = 9
5
71
40 Solve the equation.
Answer
2x + 3 = 4 3
72
41 Which value of x is the solution of the equation ?
A
B
C
D
Answer
From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011
73
Answer
42 Solve: 2 x + 5 = 7 12
6
3
74
Answer
43 Solve
75
Answer
44 Solve
76
Answer
45 Solve
77
Answer
46 Solve
78
Equations with the Same Variable on Both Sides
Math Practice
This less
Additiona
How cou
What op
(MP1)
What do
operation
(MP7)
Return to Table of Contents
79
Variables on Both Sides
Now, we will be given an equation with the same variable on both sides. These equations will look similar to the following:
These require one additional step to get all the terms with that variable to one side or the other. It doesn't matter which side you choose to move the variables to, but it’s typically most helpful to choose the side in which the coefficient of the variable will remain positive.
Math Practice
Previously, you solved equations with variables on one side, similar to the following:
MP6: Att
Emphasi
operation
80
Meaning of Solutions
Solve for x:
When you have finished solving, discuss the meaning of your answer with your neighbor.
Teacher Notes
Before we encounter the new equations, let's practice how to solve an equation with the variable on only one side.
Student
solving th
they need m
use the 8th
Have stu
(that valu
81
Meaning of Solutions
Remember that you always have the ability to check your answers by substituting the value you solved for back in to the original equation. It isn't necessary to show on each problem, but is encouraged if you feel unsure about your answer.
82
Variables on Both Sides
Answer
Which side do you think would be easiest to move the variables to?
83
Variables on Both Sides
Answer
Now, solve the equation.
84
Variables on Both Sides
Answer
Which side do you think would be easiest to move the variables to?
The left simplifyin
equation
terms), y
equation
6r 85
Variables on Both Sides
Answer
Now, solve the equation.
86
Variables on Both Sides
Example:
What do you think about this equation? What is the value of x?
Answer
When solvin
eliminated equation, th
no value 87
Variables on Both Sides
Example:
What do you think about this equation? What is the value of x?
Answer
When solvin
eliminated,
equation, value of x
88
Answer
47 Solve for f:
89
Answer
48 Solve for h: 90
Answer
49 Solve for x: 91
No Solution
Sometimes, you get an interesting answer.
What do you think about this?
What is the value of x?
3x ­ 1 = 3x + 1
­3x
­3x
­1 = +1
Since the equation is false, there is no solution! No value will make this equation true.
92
Identity
How about this one?
What do you think about this?
What is the value of x?
3(x ­ 1) = 3x ­ 3
3x ­ 3 = 3x ­ 3
­3x
­3x
­3 = ­3
Since the equation is true, there are infinitely many solutions! The equation is called an identity. Any value will make this equation true.
93
50 Solve for r:
A r = 0
C infinitely many solutions (identity)
D no solution
Answer
B r = 2
94
51 Solve for w:
B w = ­1 C infinitely many solutions (identity)
Answer
A w = ­8 D no solution
95
52 Solve for x:
A x = 0
C infinitely many solutions (identity)
Answer
B x = 24
D no solution
96
53 Solve for m:
B m = 1
C infinitely many solutions (identity)
Answer
A m = 2.2 D no solutions
97
54 In the equation
n is equal to?
C
D
Answer
A
B
From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
98
M
A
4x + 4
5x – 3
O
3x
N
17
B
Answer
55 In the accompanying diagram, the perimeter of ∆MNO is equal to the perimeter of square ABCD. If the sides of the triangle are represented by 4x + 4, 5x ­ 3, and 17, and one side of the square is represented by 3x, find the length of a side of the square.
D
C From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
99
56 Solve this equation for x.
Answer
0.5(5 ­ 7x) = 8 ­ (4x + 6)
From PARCC sample test ­ PBA non calc #2
100
57 Solve for x.
Answer
9(3 ­ 2x) = 2(10 ­ 8x)
From PARCC sample test ­ EOY non­calc #1
101
58 Solve for x. Answer
From PARCC sample test ­ EOY non­calc #18
102
RECAP
• When solving an equation with variables on both sides, choose a side to move all of them to, then continue working to isolate the variable.
• When solving an equation where all variables are eliminated and the remaining equation is false, there is No Solution.
• When solving an equation where all variables are eliminated and the remaining equation is true, there are Infinite Solutions.
103
Comparing Expressions with the Same Variable
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104
Expressions with the Same Variable
Recall that an expression is a mathematical phrase that contains ordinary numbers, variables and operations. Also, an equation is a statement indicating that two expressions are equal. In the last lesson, you were solving an equation that contained variables on both sides of the equation. In some cases, you were able to find a solution. In others, there was either no solution, or infinitely many solutions. What if you were given two expressions and trying to determine the relationship between them, instead. Could it be done?
105
Expressions with the Same Variable
You can easily compare expressions to one another. Take this example:
What is the relationship between the expressions (x + 4) and (x + 8)?
Since we don't know the relationship between the expressions yet, we can write an equation, but replace the "=" sign with a question mark.
x + 4 ? x + 8
106
Expressions with the Same Variable
What is the relationship between the expressions (x + 4) and (x + 8)?
x + 4 ? x + 8
Treat this like an equation and try to solve it by first subtracting x from both sides. Now, we have 4 ? 8
We can conclude that 4 < 8. Therefore, no matter what the value of x is, (x + 4) will always be less than (x + 8).
107
Expressions with the Same Variable
Let's try another example:
What is the relationship between the expressions 4x and ­6x?
Since we don't know the relationship between the expressions yet, we can write an equation, but replace the "=" sign with a question mark.
4x ? ­6x
108
Expressions with the Same Variable
What is the relationship between the expressions 4x and ­6x?
4x ? ­6x
Treat this like an equation and try to solve it by first adding 6x to both sides. Then, we have 10x ? 0
We can finish solving by dividing both sides by 10. The result will be
x ? 0
109
Expressions with the Same Variable
The q
MP.3: crit
x ? 0
In the expression above, there is no set value for x. Can you determine the different answers and provide evidence to support them?
Math Practice
What is the relationship between the expressions 4x and ­6x?
Ask: How
What m
Discuss.
What 110
Expressions with the Same Variable
What is the relationship between the expressions 4x and ­6x?
x ? 0
Here are the cases that your groups should have come up with. Case 1: If x > 0, then 4x > ­6x e.g. if x = 2, 4(2) > ­6(2), or 8 > ­12 Case 2: If x < 0, then 4x < ­6x
e.g. if x = ­1, then 4(­1) < ­6(­1), or ­4 < 6 Case 3: If x = 0, 4x = ­6x
e.g. if x = 0, then 4(0) = ­6(0), or 0 = 0
111
59 What is the relationship between the expressions (x ­ 10) and (x ­ 15)?
B x ­ 10 > x ­ 15
C x ­ 10 = x ­ 15
Answer
A x ­ 10 < x ­ 15
D all of the above, depending on the value of x
112
60 What is the relationship between the expressions x x A <
­3
5
x x B >
­3 5
x x C =
­3
5
Answer
x x and ?
­3
5
D all of the above, depending on the value of x
113
61 What is the relationship between the expressions (4x ­ 3) and (4x + 3)?
B 4x ­ 3 > 4x + 3
C 4x ­ 3 = 4x + 3
Answer
A 4x ­ 3 < 4x + 3
D all of the above, depending on the value of x
114
62 What is the relationship between the expressions 1 A 3x ­ 7 < (9x ­ 21)
3
1 B 3x ­ 7 > (9x ­ 21)
3
1 C 3x ­ 7 = (9x ­ 21)
3
Answer
1 3x ­ 7 and (9x ­ 21)?
3
D all of the above, depending on the value of x
115
Expressions with the Same Variable
Summer is considering the expressions 1 2(3x + 20) and (6x ­ 12) + 4(x + 11)
3
Part A
She claims that the first expression is greater than the second expression for all values of x. Is Summer correct? Explain your answer.
Part B
Write another expression that is always less than both of these expressions.
116
Expressions with the Same Variable
Summer is considering the expressions 1 2(3x + 20) and (6x ­ 12) + 4(x + 11)
3
Part A
She claims that the first expression is greater than the second expression for all values of x. Is Summer correct? Explain your answer.
If we set up our expressions with a question mark in between them, and simplify them, we would have
1 2(3x + 20) ? (6x ­ 12) + 4(x + 11)
3
6x + 40 ? 2x ­ 4 + 4x + 44
6x + 40 ? 6x + 40
117
Expressions with the Same Variable
Summer is considering the expressions 1 2(3x + 20) and (6x ­ 12) + 4(x + 11)
3
Part A
She claims that the first expression is greater than the second expression for all values of x. Is Summer correct? Explain your answer.
6x + 40 ? 6x + 40
Clearly, we can replace the "?" with an "=". Therefore, Summer is incorrect.
118
Expressions with the Same Variable
Summer is considering the expressions 1 2(3x + 20) and (6x ­ 12) + 4(x + 11)
3
Part B
Write another expression that is always less than both of these expressions.
Since both of them simplified to 6x + 40, we can write any algebraic expression that simplifies to 6x and any number less than 40. Here are 2 sample answers:
6x + 20
1 (12x ­ 10) = 6x ­ 5
2
119
63 Bobby is considering the expressions
Yes F
Answer
He claims that the first expression is less than the second. Is he correct? Explain your answer.
Secon
­0.5x No 120
64 Write an expression that is always greater than the two expressions. Answer
When you finish writing your expression, enter the number "1" into your responder
Answ
answe
­0.5 a
121
65 Debbie is considering the expressions
She claims that the two expressions are equal. Is she correct? Explain your answer.
Yes No Answer
First:
Secon
­0.
122
When you finish writing your expression, enter the number "1" into your responder.
Answer
66 Write an expression that is always less than the two expressions.
Answ
answe
­0.8 an
123
67 Martin is considering the expressions
He wants to know if one expression is greater than the other for all values of x.
Part A Which statement about the relationship between the expressions is true?
A The value of the expression is always equal to B The value of the expression is always less than the value of the expression .
C The value of the expression is always greater Answer
the value of the expression .
than the value of the expression .
D The value of the expression is sometimes greater than and sometimes less than to the value of the expression .
From PARCC ­ PBA calculator #8
124
68 Martin is considering the expressions
He wants to know if one expression is greater than the other for all values of x.
Answer
Part B
Show or explain how you found your answer to Part A. When you finish, type in "1" into your Responder.
3.5x + 3.5
From PARCC sample test ­ PBA ­ calculator #8
125
69 Martin is considering the expressions
He wants to know if one expression is greater than the other for all values of x.
Answer
Part C
Write a new expression that always has a greater value than both of these expressions. Enter "1" into your responder when you finish.
Answ
answer
3.5 an
From PARCC sample test ­ PBA ­calculator #8
126
Writing and Solving Algebraic Equations
Return to Table of Contents
127
Recall from 6th grade and 7th grade, we translated phrases/
sentences into expressions and equations. Now, we can extend this understanding to writing and solving algebraic equations, and using them to solve word problems.
Let's start off by reviewing key words for our operations. What are some of them?
Answer
Writing and Solving Algebraic Equations
128
Review of Writing and Solving Algebraic Equations
The following formative assessment questions are review from 7th grade. If further instruction is need, see the presentation at:
https://www.njctl.org/courses/math/7th­grade/
equations­inequalities­7th­grade/
129
70 Which equation can be made from the sentence below:
The difference between 15 and a number is 18.
B 18 ­ n = 15
C n ­ 15 = 18
Answer
A n ­ 18 = 15
D 15 ­ n = 18
130
Answer
71 What is the solution to the equation that represents the sentence below?
The difference between 15 and a number is 18.
131
n A ­ 12 = ­3
4
n B + 12 = ­3
4
n C ­ 12 = 4
­3
n D + 12 = 4
­3
Answer
72 Which equation can be made from the sentence below:
Twelve more than the quotient of a number and four is ­3.
132
Answer
73 What is the solution to the equation that represents the sentence below?
Twelve more than the quotient of a number and four is ­3.
133
2 A n + (­1) = 7
5
2 B n ­ (­1) = 7
5
2 C n + 7 = ­1
5
2 D n ­ 7 = ­1
5
Answer
74 Which equation can be made from the sentence below:
Two fifths of a number increased by 7 is ­1.
134
Answer
75 What is the solution to the equation that represents the sentence below?
Two fifths of a number increased by 7 is ­1.
135
76 Translate and solve. Answer
10 equals the sum of four and the quotient of a number n and 12.
136
77 Translate, then solve the equation.
Answer
Four more than three times a number is 13.
137
78 Translate, then solve the equation.
Answer
How old am I if 400 reduced by 2 times my age is 342?
138
79 Translate and solve. Answer
Twice the quantity of x plus eight is negative 2.
139
80 Translate and solve. Answer
Negative seven times the sum of eighteen and a number s is equal to 49.
140
81 Translate and solve. Answer
132 is equal to negative twelve times the quantity of the sum of a number x and eighteen.
1
141
82 Translate and solve. Answer
Negative one is the same as the quotient of the sum of four and x, and six.
142
Writing and Solving Algebraic Equations
The a
We can use a variable to show an unknown.
A constant will be any fixed amount.
If there are two separate unknowns, relate one to the other. Math Practice
We can use our algebraic translating skills to solve other problems. Ask: W
What d
How ca
sy
What c
this p
143
Writing and Solving Algebraic Equations
The school cafeteria sold 225 chicken meals today. They sold twice the number of grilled chicken sandwiches than chicken tenders. How many of each were sold? 2c + c = 225
chicken
sandwiches
chicken
tenders
total meals
c + 2c = 225
3c = 225 3 3
c = 75
The cafeteria sold 150 grilled chicken sandwiches and 75 tenders. 144
Writing and Solving Algebraic Equations
Many times with equations there will be one number that will be the same no matter what (constant) and one that can be changed based on the problem (variable and coefficient). Example: George is buying video games online. The cost of the video is $30.00 per game and shipping is a flat fee of $7.00. He spent a total of $127.00. How many games did he buy in all?
145
Writing and Solving Algebraic Equations
George is buying video games online. The cost of the video is $30.00 per game and shipping is a flat fee of $7.00. He spent a total of $127.00. How many games did he buy in all?
Notice that the video games are "per game" so that means there could be many different amounts of games and therefore many different prices. This is shown by writing the amount for one game next to a variable to indicate any number of games. 30g
cost of
one video
game
number
of games
146
Writing and Solving Algebraic Equations
George is buying video games online. The cost of the video is $30.00 per game and shipping is a flat fee of $7.00. He spent a total of $127.00. How many games did he buy in all?
Notice also that there is a specific amount that is charged no matter what, the flat fee. This will not change so it is the constant and it will be added (or subtracted) from the other part of the problem. 30g + 7
cost of
one video
game
number
of games
the cost of the shipping
147
Writing and Solving Algebraic Equations
George is buying video games online. The cost of the video is $30.00 per game and shipping is a flat fee of $7.00. He spent a total of $127.00. How many games did he buy in all?
"Total" means equal so here is how to write the rest of the equation. 30g + 7 = 127
cost of
one video
game
number
of games
the cost of
shipping
the total amount
148
Writing and Solving Algebraic Equations
George is buying video games online. The cost of the video is $30.00 per game and shipping is a flat fee of $7.00. He spent a total of $127.00. How many games did he buy in all?
Answer
Now you can solve it.
149
Writing and Solving Algebraic Equations
5
Answer
Walter is a waiter at the Towne Diner. He earns a daily wage of $50, plus tips that are equal to 15% of the total cost of the dinners he serves. What was the total cost of the dinners he served if he earned $170 on Tuesday?
From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
150
Answer
83 Lorena has a garden and wants to put a gate to her fence directly in the middle of one side. The whole 1
length of the fence is 24 feet. If the gate is 4 feet, 2
how many feet should be on either side of the fence?
151
A
B
C
D
12p + 27 = 147 12p + 27p = 147 27p + 12 = 147 39p = 147 Answer
84 Lewis wants to go to the amusement park with his family. The cost is $12.00 for parking plus $27.00 per person to enter the park. Lewis and his family spent $147. Which equation shows this problem?
152
Answer
85 Lewis wants to go to the amusement park with his family. The cost is $12.00 for parking plus $27.00 per person to enter the park. Lewis and his family spent $147. How many people went to the amusement park WITH Lewis?
153
A
B
C
D
9 + 68 = 239
9d + 68 = 239
68d + 9 = 239
77d = 239
Answer
86 Mary is saving up for a new bicycle that is $239. She has $68.00 already saved. If she wants to put away $9.00 per week, how many weeks will it take to save enough for her bicycle? Which equation represents the situation?
154
Answer
87 Mary is saving up for a new bicycle that is $239. She has $68.00 already saved. If she wants to put away $9.00 per week, how many weeks will it take to save enough for her bicycle? 155
88 You are selling t­shirts for $15 each as a fundraiser. You sold 17 less today then you did yesterday. Altogether you have raised $675. Be prepared to show your equation! Answer
Write and solve an equation to determine the number of t­shirts you sold today.
156
89 Rachel bought $12.53 worth of school supplies. She still needs to buy pens which are $2.49 per pack. She has a total of $20.00 to spend on school supplies. How many packs of pens can she buy?
Be prepared to show your equation!
Answer
Write and solve an equation to determine the number of packs of pens Rachel can buy.
157
90 The product of ­4 and the sum of 7 more than a number is ­96.
Be prepared to show your equation! Answer
Write and solve an equation to determine the number.
158
Write and solve an equation to determine the number of subscribers they had each year.
Be prepared to show your equation!
How many subscribers last year? Answer
91 A magazine company has 2,100 more subscribers this year than last year. Their magazine sells for $182 per year. Their combined income from last year and this year is $2,566,200. 182x + x = 600
6000 su
8100 su
159
Teachers:
Use this Mathematical Practice Pull Tab for the next 2 SMART Response slides.
Math Practice
The add
Ask: W
How can
sy
Could What c
this p
160
Write and solve an equation to determine the width of the rectangle.
Be prepared to show your equation! Answer
92 The length of a rectangle is 9 cm greater than its width and its perimeter is 82 cm.
161
Write and solve an equation to determine the length of a side of the hexagon.
Be prepared to show your equation! Answer
93 The perimeter of a regular hexagon is 13.2 cm. 162
Answer
94 Two angles are complementary. One angle has a measure that is five times the measure of the other angle. What is the measure, in degrees, of the larger angle?
From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
163
Answer
95 Two angles are supplementary. One angle has a measure that is 40 degrees larger than the measure of the other angle. What is the measure, in degrees, of the larger angle?
x = 7
From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
164
96 Translate, then solve the equation.
Answer
Together two laptops cost $756. Laptop #1 is one third the cost of laptop #2. What is the cost of the cheaper laptop?
165
97 Filipo is building a rectangular sandbox for his younger brother. The length of the sandbox is 1 foot longer than twice the width of the sandbox. The perimeter of the sandbox is 29 feet. A w + w + 2 = 29
B w + 2w + l = 29
Answer
Part A Which equation could be used to determine w, the width, in feet, of the sandbox?
C 2w + 2(w + 2) = 29
D 2w + 2(2w + l) = 29
From PARCC sample test ­ EOY calculator #2
166
98 Part B (Continued from previous question) Answer
What is the width, in feet, of the sandbox?
From PARCC sample test ­ EOY calculator #2
167
Translating & Solving Consecutive Integer Problems
Return to Table of Contents
168
Consecutive Integers
Another type of word problem that you might encounter involve consecutive integers.
Consecutive integers are integers that are in a row and have a difference of 1. For example
1, 2, 3 ­6, ­5, ­4 99, 100, 101
If we need to solve a consecutive integer word problem, it is necessary to assign a variable to each unknown and then set them equal to the given value.
Let's see an example...
169
Consecutive Integer Problems
The sum of two consecutive integers is 57. Find the two integers.
Begin by assigning the 1st integer as x and the 2nd integer as x + 1.
Then add both integer representations and set the expression equal to the number given.
x + x + 1 = 57 solve the equation
2x + 1 = 57
2x = 56
x = 28 once you solve for x, then substitute the value in to find the value of the next integer.
x + 1 = 29
The integers are 28 & 29
170
Consecutive Integer Problems
Try the examples below.
The sum of two consecutive integers is ­33. Find the integers.
1st integer = x
2nd integer = x+1
x + x + 1 = ­33
2x + 1 = ­33
2x = ­34
x = ­17
x + 1 = ­16
slide to reveal
The sum of two consecutive integers is 205. Find the integers.
1st integer = x
2nd integer = x+1
x + x + 1 = 205
2x + 1 = 205
2x = 204
x = 102
x + 1 = 103
slide to reveal
171
Consecutive Integer Problems
1st integer = x
2nd integer = x+1
3rd integer = x+2
x+x+1+x+2+93
3x + 3 = 93
3x = 90
x = 30
x + 1 = 31
x + 2 = 32
slide to reveal
The sum of two consecutive even integers is 114. Find the integers.
1st integer = x
2nd integer = x+2
x + x + 2 = 114
2x + 2 = 114
2x = 112
x = 56
x + 2 = 58
Hint &
Math Practice
The sum of three consecutive integers is 93. Find the integers.
The 2
ad
Ask: W
make
Can you
symb
slide to reveal
172
Answer
99 The sum of two consecutive integers is 35. Find the larger of the two.
173
Answer
100 The sum of two consecutive integers is 201. Find the smaller of the two integers.
174
Answer
101 The sum of three consecutive integers is 615. Find the middle one.
175
Answer
102 The sum of two consecutive odd integers is 624. Find the smaller integer.
176
Teacher Notes
Glossary & Standards
Vocabu
in the p
box the
linked of the p
word d
Return to
Table of
Contents
177
Consecutive Integers
Integers that are in a row and have a difference of 1
1, 2, 3
­9, ­8, ­7
x, x + 1, x + 2
Back to Instruction
178
Distributive Property
For all real numbers a, b, c, a(b + c) = ab + ac and a(b ­ c) = ab ­ ac. a(b + c) = ab + ac
a(b ­ c) = ab ­ ac
3(2 + 4) = (3)(2) + (3)(4) =
6 + 12 = 18
3(x + 4) = 48 (3)(x) + (3)(4) = 48 3(2 ­ 4) = (3)(2) ­ (3)(4) =
6 ­ 12 = ­6
3x = 36 x = 12 3x + 12 = 48 Back to Instruction
179
Equation
A mathematical statement, in symbols, where two things are exactly the same (or equivalent).
1 + 2 = 3
22 = 20 + 2
4 + 9 = 13
7x = 28
(where x = 4)
3y + 2 = 14
(where y = 4)
14 ­ 1 = 3z + (where z = 4)
1
7 + 4 = 90
5 = ­3 + 10
3x + 6 = 11
(where x = 4)
Back to Instruction
180
Expression
Numbers, symbols and operators (such as + and ×) grouped together that show the value of something.
7x
3y + 2
2 ­ 9b
7 x 6
­0.5a
7x = 21
11 = 3y + 2
11 ­ 1 = 3z + 1
Remember!
7x "7 times x"
"7 divided by x"
Back to Instruction
181
Identity An equation that has infinitely many solutions. 3(x ­ 1) = 3x ­ 3
3x ­ 3 = 3x ­ 3
­3x ­3x
­3 = ­3
7(2x + 1) = 14x + 7
14x + 7 = 14x + 7
­14x ­14x
7 = 7
3x ­ 1 = 3x + 1
­3x ­3x
­1 = +1
Back to Instruction
182
Inverse Operation
The operation that reverses the effect of another operation.
Addition
_
+
Subtraction
Multiplication
x
÷
Division
­ 5 + x = 5 + 5
+ 5
x = 10
11 = 3y + 2
­ 2
­ 2
9 = 3y
÷ 3 ÷ 3
3 = y
Back to Instruction
183
Like Terms Terms whose variables (and their exponents) are the same.
3x x3 27x3
x 1/2x
­2x3
3
15.7x 1/4x
5x
­5x3
2.7x3
­2.3x
5x3
5
NOT 5x2
LIKE TERMS! 5x
5x4
Back to Instruction
184
No Solution
An equation that is false.
3x ­ 1 = 3x + 1
­3x ­3x
­1 = +1
8x ­ 4 = 8x + 6
­8x ­8x
­4 = 6
3(x ­ 1) = 3x ­ 3
3x ­ 3 = 3x ­ 3
­3x ­3x
­3 = ­3
Back to Instruction
185
Two­Step Equation
An equation that contains two operations
3x ­ 1 = 11
x + 8 = 6
3
3(x ­ 1) + x = 25
Back to Instruction
186
Variable
A symbol for a number that is unknown. any letter towards end of alphabet!
x y z
u v
7x = 21
*Sometimes Greek!
11 = 3y + 2
11 ­ 1 = 3z + 1
Back to Instruction
187
Standards for Mathematical Practices
MP1 Make sense of problems and persevere in solving them.
MP2 Reason abstractly and quantitatively.
MP3 Construct viable arguments and critique the reasoning of others.
MP4 Model with mathematics.
MP5 Use appropriate tools strategically.
MP6 Attend to precision.
MP7 Look for and make use of structure.
MP8 Look for and express regularity in repeated reasoning.
Click on each standard to bring you to an example of how to meet this standard within the unit. 188