Download Lesson 7.1 Writing One Variable Equations Name

Lesson 7.1
Writing One Variable Equations
Word Sentence
The sum of a number n and 7 is 15.
A number y decreased by 4 is 3.
12 times a number p equals 48.
A number less than twenty is four.
Twenty less than a number is twelve.
The quotient of a number m and 4 is 12.
Name: __________________
Equation
n + 7 = 15
y–4=3
12p = 48
20 – n = 4
n – 20 = 12
m = 12
4
PRACTICE: Page 298
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Write a word sentence for this equation:
35 – a = 13
_____________________________________________________________
21) NOTE – Either a or b is a yes. Figure out which one is a yes, and then see if
you can solve it.
22) Find the length of side s.
Lesson 7.2
Solving Equations Using
Addition or Subtraction
Name: ___________________________
Equations that have either addition or subtraction can be solved using addition or
subtraction.
Equation
29 + n = 37
n + 13 = 40
n – 15 = 45
24 – n = 13
To Solve
37 – 29 = 8
40 – 13 = 27
15 + 45 = 60
24 – 13 = 11
Solution
n=8
n = 27
n = 60
n = 11
Check
29 + 8 = 37
27 + 13 = 40
60 – 15 = 45
24 – 11 = 13
*NOTE – For both addition equations, you use subtraction to solve. But for the two
subtraction equations – one uses addition, and the other uses subtraction. This can
be confusing.
Mr. Binder’s suggestion: First ask yourself if the solution is either LARGER or
SMALLER than the biggest number in the equation. If it’s Larger – then use
addition. If it’s smaller = then use subtraction.
Example 1: n – 30 = 50
The solution is LARGER than 50 – so use addition.
30 + 50 = 80, so n = 80
Example 2: 44 – n = 11
To check: 80 – 30 = 50
The solution is SMALLER than 44 – so use subtraction.
44 – 11 = 33, so n = 33
To check: 44 – 33 = 11
PRACTICE: PAGE 305
#6 to 11
(Print YES or NO, and if NO – print the actual solution)
6)
7)
10)
11)
8)
9)
#18 to 29
(Solve the equation and just print your solution)
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19)
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21)
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24)
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27)
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33)
34)
CHALLENGE PROBLEM
Find the LARGEST 6-digit whole number with this one condition:
• The range of the six digits is three times the mean of the six digits
_____ _____ _____ , _____ _____ _____
Lesson 7.3
Solving Equations using
Multiplication or Division
Name: ____________________
Solving Multiplication Equations
3n = 21
Use Division: 21 divided by 3 = 7
Check: 3 x 7 = 21
Solving Division Equations with Multiplication
n = 3
Use Multiplication: 8 x 3 = 24
8
Check:
24 = 3
8
Solving Division Equations with Division
30 = 3
Use Division: 30 divided by 3 = 10
n
Check:
30 = 3
10
Solving 2-Step Equations
4n = 12
Use Multiplication: 3 x 12 = 36, so 4n = 36
3
Use Division: 36 divided by 4 = 9
Check: 4x9=36,
PRACTICE: PAGE 312
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OVER
36 = 12
3
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30)
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33)
34)
35)
CHALLENGE PROBLEM:
Find the LARGEST 6-digit whole number with these conditions:
• The difference of the middle two digits is half the sum of the outer two
digits
• No single digit can be used more than once
• The sum of all digits is less than 27
______
______
______ , ______
______
______
Lesson 7.4
Writing Equations in Two Variables
Name: __________________
Independent and Dependent Variables
A public radio station is accepting donations of money. A local company has
promised to donate twice the amount of all donations given.
Look at this equation: y = 2x Let x represent the amount of money that is
donated. Let y represent the amount that the company will donate.
x is considered the independent variable, because it is not known how much money
will be donated. y is considered the dependent variable, because the value of y is
dependent on the value of x.
Identifying Solutions of Equations in Two Variables
y = 2x
An ordered pair is a solution for an equation in two variables.
(3, 6) is one solution for the equation; y = 2x.
The 3 represents the value of x, and the 6 represents the value of y.
If x = 3, then y = 6, so the solution (3, 6) works.
(0, 0) is another solution that works, since if x = 0, then y = 0.
(4, 7) is an incorrect solution, because if x = 4, the y = 8 (not 7).
Using Equations in Two Variables
The equation: y = 128 – 8x gives the amount (y) of ounces of milk remaining in a
gallon jug after you pour x cups.
The x is the independent variable, because the amount of milk poured can be
different. The y is the dependent variable, because it is dependent on x (the
amount poured).
How much milk remains after 10 cups are poured?
Substitute 10 in place of x:
y = 128 – 8 • 10
Solve for y: 128 – 80 = 48, so y = 48
Over
PAGE 319
Perimeter of a Rectangle: P = 2l + 2w
Area of a Trapezoid: .5h(base 1 + base 2)
4)
5)
For # 6 to 11, print YES or NO. If NO, find one ordered pair that IS a solution.
6)
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For #13 to 17, print the equation, then tell which letter is the dependent variable
and which is the independent variable. (Also, for #17 – answer the question, too.)
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For # 18 to 21, print your own Independent or Dependent Variable
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38) a:
b:
Lesson 7.5 Writing and Graphing Inequalities
Name: __________________
Inequality Symbols
> greater than
< less than
> greater than or equal to
< less than or equal to
Math Sentences
A number c is less than -4
A number k plus 5 is greather than or equal to 8
4 times a number q is at most 16
Tell whether a given number is
n>4
Is 5 a solution?
n<5
Is -2 a solution?
n<5
Is 6 a solution?
solution.
Inequalities
c < -4
k+5>8
4q < 16
a solution to an inequality
5 is greater than 4, so 5 is a solution.
-2 is less than 5, so -2 is a solution.
6 is not less than or equal to 5, so 6 is not a
Graphing Inequalities
-5
-4
-3
graph n > 2
-5
-4
-2
-1
0
1
2
3
4
5
*Note: the open circle indicates 2 is NOT included
-3
graph n < 4
-2
-1
0
1
2
3
4
5
*Note: the filled in circle indicates 4 IS included
PRACTICE: Page 329
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For #11 to 16, print yes or no
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For #17 to 20, print a, b, c, or d
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For #21 to 24, write an inequality only
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-5
-4
-3
-2
-1
0
1
2
3
4
5
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5
-4
-3
-2
-1
0
1
2
3
4
5
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Lesson 7.6 Solving Inequalities using
Name:
________________________
Addition or Subtraction
Addition Inequalities
Addition inequalities will ALWAYS be solved using subtraction.
Examples
Subtract
Solution
3+n>9
9–3=6
n>6
5 + n < 12
12 – 5 = 7
n<7
Subtraction Inequalities
Subtraction inequalities WHEN THE VARIBLE COMES FIRST can be solved
using addition.
Examples
n – 5 > 12
n – 20 < 30
Add
5 + 12 = 17
20 + 30 = 50
Solution
n > 17
n < 50
PRACTICE: Page 336
For Problems #5 to 16, just print the solution – you don’t need to graph
them.
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Try Solving These Inequalities
1) 31 – n > 16
______
2) 40 – n < 20
______
3) 16 < n + 13
______
4) -4n > 20
______
Find the LARGEST 6-digit whole number in which the mean of the six digits is
twice the range of the six digits.
_____ _____ _____ _____ _____ _____
Lesson 7.7
Solving Inequalities using
Name: ______________________
Multiplication or Division
Solving Multiplication Inequalities
To solve a multiplication inequality, always use DIVISION.
Examples
Divide
Solution
4n > 24
24 ÷ 4
n>6
12n < 60
60 ÷ 12
n<5
3n > 12
12 ÷ 3
n > 16
4
4
Solving Division Inequalities (When the variable is on top only)
To solve a division inequality when the variable is on top, always use
MULTIPLICATION.
Examples
Multiply
Solution
n>4
5x4
n > 20
5
n<8
10
10 x 8
n < 80
PRACTICE: Page 342
For # 6 to 21, just solve the inequality (don’t graph them)
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For #28 and 29: Solve only (don’t graph them) You have to show what x can be to
satisfy both equations in each problem
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7.1 HW
Name: _______________________________
Write an equation for each math sentence below. (You don’t need to solve them)
Use n as your variable.
Statement
Equation
1) A number and seven is fifteen.
2) Six times a number is thirty-six.
3) A number divided by eight is four.
4) Seventy divided by a number is seven.
5) Eight less than a number is twenty.
6) A number less than eight is three.
7) If you add a number to thirteen you’ll get forty.
8) Nine times a number is ninety-nine.
Write your own math sentence for each equation below:
9) 45 – n = 31
______________________________________________
10) 57 + n = 100
______________________________________________
7.2 HW
Name:
_________________________________________
Solve each equation below. You may use a calculator.
1) 28 – n = 12
n = ____
2) 42 + n = 63
n = ____
3) n – 17 = 25
n = ____
4) n + 18 = 40
n = ____
5) 41 – n = 11
n = ____
6) 44 = n + 17
n = ____
7) n – 21 = 41
n = ____
8) 49 = 28 + n
n = ____
9) 59 – n = 0
n = ____
10) n + 33 = 93
n = ____
11) n – 10 = 40
n = ____
12) 105 + n = 200
n = ____
7.3 HW
Name: ___________________________________
Solve each equation below. YOU MAY USE A CALCULATOR.
1)
4n = 44
n = ____
2)
n =5
12
n = ____
3)
15 = 5
n
n = ____
4)
56 = 7n
n = ____
5)
n= 7
10
n = ____
6)
32 = 8
n
n = ____
7)
6n = 6
5
n = ____
8)
12n = 24
2
n = ____
9)
20n = 20
3
n = ____
10)
40n = 10 n = ____
12
7.4 HW
Name: _____________________________________
For each situation below, indicate which variable is dependent and independent:
1) For every dollar donated by the students (d), the staff will donate 3 dollars.
s = 3d
___ is dependent
___ is independent
2) Joe’s income (i) is based on how many hours (h) he works.
___ is dependent
___ is independent
3) Joe has $100. He buys some cans of soup (s). Each can costs $5. After he pays
with the $100, his change (c) will be based on this equation: c = 100 – 5s
___ is dependent
___ is independent
Circle all correct solutions for each equation. It is possible that there are no
solutions given, and possible there is more than one solution.
4) y = 3 + x
(2, 5)
(0, 3)
(5, 7)
5) y = x – 1
(2, 1)
(5, 3)
(3, 0)
6) y = 2x – 2
(4, 6)
(1, 0)
(3, 4)
7) y = 5x
(0, 1)
(2, 5)
(3, 8)
8) y = 3 – x
(2, 1)
(0, 3)
(3, 0)
Use the situation and equation in problem #3 above: c = 100 – 5s
Solve the following problems based on this equation:
9) How much change will Joe get if he buys 12 cans of soup?
$ _____
10) How much change will Joe get if he buys 5 cans of soup?
$ _____
11) How many cans can Joe buy if he spends all his money on soup?
_____ cans
12) If the soup is on sale for half price, and Joe buys 30 cans, how much change will
he get?
$ _____
7.5 HW
Name: __________________________________________
> greater than
> greater than or equal to
< less than
< less than or equal to
Print an inequality for each math sentence below:
1) A number is greater than or equal to 5
__________________
2) 4 times a number is less than 17
__________________
3) 2 less than 5 times a number is less than 24
__________________
Print Y for yes or N for no to indicate whether the given solution is a correct
solution to each inequality below:
4) n > 4
n=6
___
5) 2n < 8
n=4
___
n=6
___
6) 3(n + 3) > 20
n=4
___
7) 12 > 2
n
8) 5n – 1 < 19
n=4
___
9) 2n + 2 > 5
n=1
___
Graph each inequality below. Remember, an open circle DOES NOT include the
number on the number line, and a closed circle DOES.
10) n > 2
-5
-4
-3
-2
-1
0
1
2
3
4
5
-3
-2
-1
0
1
2
3
4
5
-3
-2
-1
0
1
2
3
4
5
11) n < -2
-5
-4
12) n > -3
-5
-4
7.6 HW
Name: ____________________________________
Solve each inequality below:
1) n + 13 > 20 ______
2) 22 + n < 35 ______
3) n – 15 > 18 ______
4) n – 9 < 9
5) 27 + n < 40 ______
6) n + 11 > 14 ______
7) n – 22 > 50 ______
8) n – 1 < 39 ______
9) n + 29 < 39 ______
10) 31 + n > 55______
11) n – 24 < 44
12) n – 2 < 23 ______
______
______
7.7 HW
Name: ____________________________________________
Solve each inequality below. YOU MAY USE A CALCULATOR.
1)
7n < 56
_______
2)
13n > 52
_______
3)
n < 12
7
_______
4)
n>5
15
_______
5)
4n > 30
5
_______
6)
2n < 45
5
_______
7)
n>2
15
_______
8)
n < 25
8
_______
9)
22n > 66
_______
10)
100n < 500 _______
CHALLENGE PROBLEM:
Find the LARGEST 6-digit whole number with these conditions:
• The sum of the composite digits is twice the sum of the prime
digits
• No digit is used more than once
_______ _______ _______ , _______ _______ _______