Reproduce EQ3 EQUATIONS AND INEQUALITIES STUDENT PACKET 3: SOLVING EQUATION 3

Transcription

Name ___________________________
Period __________
Date ___________
EQ3
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EQUATIONS AND INEQUALITIES
STUDENT PACKET 3: SOLVING EQUATION 3
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STUDENT PACKET
Solving Equations With Rational Coefficients
• Solve multi-step algebraic equations.
• Solve equations that involve non-integer values.
• Justify the steps in equation solving.
• Solve equations for one variable in terms of the others.
1
EQ3.2
Geometry and Number Problems Revisited
• Use diagrams to organize information.
• Use algebra to solve problems.
6
EQ3.3
Coin Problems
• Solve mixture problems that involve money.
10
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EQ3.1
Vocabulary, Skill Builders, and Review
17
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EQ3.4
Equations and Inequalities (Student Pages)
EQ3 – SP
Solving Equations 3
WORD BANK
Definition
Example or Picture
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Word
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equation
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numerical coefficient
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rational numbers
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solution to an
equation
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variable
EQ3 – SP0
Solving Equations 3
3.1 Solving Equations with Rational Coefficient
SOLVING EQUATIONS WITH RATIONAL COEFFICIENTS
Set (Goals)
We will solve more algebraic equations
using algebraic notation. We will justify
the steps used to solve equations. Some
equations will involve non-integer values.
• Solve multi-step algebraic equations.
• Solve equations that involve non-integer
values.
• Justify the steps in equation solving.
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Ready (Summary)
Compute.
2.
7.8 – 9.6
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1. 7.8 + 9.6
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Go (Warmup)
4. (-8.6) ÷ (4.3)
2 3
+
3 5
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3. (-8.6) • (4.3)
2 3
−
3 5
8.
3
4
1
2
-
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 3   1
7.  -  •  - 
 4  2
6.
EQ3 – SP1
Solving Equations 3
SOLVING EQUATIONS MENTALLY
5.6 = x + 2.3
2.
3.
5.5 = 3 x + 2.5
4.
5.
x+
x − 1.5 = 1.5
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-0.2 x = - 0.04
3
= 2
4
x−
8.
1
- x = 8
2
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6.
1
x = 8
2
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7.
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1
1
= 4
2
2
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1.
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Solve each equation. Use mental strategies if possible.
EQ3 – SP2
Solving Equations 3
SOLVING EQUATIONS ALGEBRAICALLY
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Solve each equation. Justify each step.
Equation
1
x +3 = 5
2
2.
2x +
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1.
What did you do?
-2.6 = 3 x + 2.5
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3.
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1
5
=
2
6
EQ3 – SP3
Solving Equations 3
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SOLVING EQUATIONS ALGEBRAICALLY
(continued)
Solve each equation. Justify each step.
Equation
What did you do?
1.6 x + 9.8 = 3.8 x + 5.4
5.
1
1
x+2 =
x −6
2
4
3x −
4
1

= 2 x + 
5
5

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6.
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4.
EQ3 – SP4
Solving Equations 3
Solve each equation.
3.
3
x + 4 = -5
10
2.
x + 2.5 = 4.6
4.
2 x + 3.6 = 3( x + 2.5)
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1.
2
4
3x +
=
5
5
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PRACTICE
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1.5 x + 3 = 2( x − 2.5)
1
x+4 = x
2
8.
-2.5 x + 3 = x − 4
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5.
6.
3
5
x+
= x+
4
6
EQ3 – SP5
Solving Equations 3
SOLVING FOR ONE VARIABLE IN TERMS OF ANOTHER
Solve for x:
-3x = 6
2a.
Solve for x:
-14 + 5 + x = 20
3a.
Solve for z:
8 + 2z = -28
1b.
Solve for h:
2a.
Solve for c:
bh = A
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1a.
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You have learned to use properties of arithmetic and equality to solve equations. We will use
the same properties to solve for one variable in terms of another.
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a + b + c = 2b
Solve for W:
4b.
Solve for y:
2L + 2W = 40
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3b.
Solve for y:
12 + 2y = 30
x + 2y = -30
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4a.
EQ3 – SP6
Solving Equations 3
3.2 Geometry and Number Problems Revisited
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GEOMETRY AND NUMBER PROBLEMS REVISITED
Ready (Summary)
Set (Goals)
Solve for x.
6x – 1 = -4(x – 6)
2.
2
3
x −2 =
x +1
3
4
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1.
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Go (Warmup)
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We will use diagrams and algebra to solve
problems that involve geometric concepts
and number concepts.
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Translate the word expressions into algebraic notation. Then simplify the expression.
3. The sum of a number and half of the number.
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4. Seven less than three times a number, minus the number.
EQ3 – SP7
Solving Equations 3
GEOMETRY AND NUMBER PROBLEMS 1
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For each problem, underline what you are trying to find. Then answer questions a-d.
2. The sum of a number and 8 times that
number is 1,107. What is the number?
a. Define the variables using words or
pictures.
pictures.
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b. Write an equation and solve.
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1. The perimeter of an isosceles triangle is
187 units. It has two congruent sides that
are 10 less than four times the length of
the third side. What is the length of each
of the congruent sides?
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c. Write the solution in words.
d. Check the solution.
Sa
EQ3 – SP8
Solving Equations 3
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2. Half of a number subtracted from twice
the number is equal to 84. What is the
number?
pictures.
pictures.
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1. The longer side of a parallelogram is 6
more than twice the length of the shorter
side. The perimeter is 108 units. What is
the length of each side?
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e. Check the solution.
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e. Check the solution.
EQ3 – SP9
Solving Equations 3
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1. A triangle has a perimeter of 30 units. The 2. The three angles in a triangle are
second side is 3 times the length of the
consecutive even numbers. Find the
1
number of degrees in each angle.
first side. The third side is 5 units more
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2
than the length of the second side. What
is the length of each side?
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pictures.
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pictures.
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f. Check the solution.
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f. Check the solution.
EQ3 – SP10
Solving Equations 3
3.3. Coin Problems
Ready (Summary)
Set (Goals)
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• Solve mixture problems involving money.
We will use pictures to solve mixture
problems involving coins.
Determine the number of cents in each amount.
2.
$3.00
3.
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$0.25
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1.
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COIN PROBLEMS
$14.50
Determine the value of each collection in cents.
5 nickels
5.
6 dimes
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4.
8. 12 quarters and 3 dimes
2 quarters
9. 2 nickels, 3 pennies, and
14 dimes
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7. 3 half-dollars
6.
Determine the number of coins in each container
11. There are 20 coins in a
jar. There are 12
nickels. The rest are
dimes. How many dimes
are there?
12. There are two coin jars.
One contains nickels and
the other contains dimes.
The total number of
coins is 20. There are n
nickels. How many
dimes are there?
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10. There are 10 coins in a
jar. There are 4 nickels.
The rest are quarters.
How many quarters are
there?
EQ3 – SP11
Solving Equations 3
3.3. Coin Problems
COIN JAR PICTURES
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Here is a way to record information about the contents of a coin jar.
Number of coins is unknown
Example: A coin jar contains 7 nickels.
Example: A coin jar contains n nickels.
•
Type of coin
•
Type of coin
•
Number of coins
•
Number of coins
•
Value of coin
•
Value of coin
•
Value of coin jar
7
5 cents
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5 cents
nickels
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nickels
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Number of coins is known
•
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Use a coin jar picture to record information about each coin collection.
•
•
Type of coin
Number of coins
•
Number of coins
Value of coin
•
Value of coin
•
Value of coin jar: 
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•
2. A coin jar contains q quarters.
•
Type of coin
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1. A coin jar contains 15 quarters.
EQ3 – SP12
Solving Equations 3
3.3. Coin Problems
Use a coin jar picture to record information about each coin collection.
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COIN JAR PICTURES (continued)
4. A coin jar contains d dimes .
•
Type of coin
•
Type of coin
•
Number of coins
•
Number of coins
•
Value of coin
•
Value of coin
•
•
Value of coin jar: 
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3. A coin jar contains 12 dimes.
•
•
Type of coin 
Number of coins
Think:
20 coins
minus the
number of
nickels results
in the
remaining
dimes
Value of coin 
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Think:
n nickels
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5. There are two coin jars. One contains nickels and the other contains dimes. The total
number of coins is 20.
Value of coin jar 
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•
EQ3 – SP13
Solving Equations 3
3.3. Coin Problems
SOLVING COIN PROBLEMS 1
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Follow the steps to solve.
A collection of 10 coins consist of all nickels and quarters. The total value of the coins is
$1.90. How many of each coin are there in the collection?
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a. Define the variables using words and pictures. Make a coin jar picture with the
information. Include the value of the collection. (define variables using words and
pictures)
Make note
of the total
value in
cents
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Type of coin 
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Number of coins
Value of coin 
____________
+ ____________ = ____________
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b. Write an equation and solve it. The equation will show that the sum of the values in
each coin jar equal the value of the collection.
c. Write the solution in words. That is, answer the question.
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There are _____ nickels and _____ quarters.
_____ cents + _____ cents = _____ cents
EQ3 – SP14
Solving Equations 3
3.3. Coin Problems
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A collection of 20 coins consists of all dimes and quarters. The total value of the coins is
$3.80. How many of each are there in the collection?
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a. Define the variables using words and pictures.
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Type of coin 
Value of coin 
____________
+
____________
= ___________
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b. Write an equation and solve it.
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Number of coins
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There are ________________________________________________.
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_____ cents + _____ cents = ____ cents
EQ3 – SP15
Solving Equations 3
3.3. Coin Problems
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A combination of 40 coins consists of all nickels and dimes. The total value of the coins is
$3.10. How many of each coin are there In the collection?
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a. Define the variables using words and pictures.
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Type of coin 
Value of coin 
____________
+
____________
= ___________
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b. Write an equation and solve it.
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Number of coins
m
Sa
EQ3 – SP16
Solving Equations 3
3.3. Coin Problems
COIN PROBLEM CHALLENGE
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Solve each coin problem.
1. A collection of coins contains dimes and nickels. There are 5 times as many dimes as
nickels. The value of the collection is $3.85. Find the number of dimes.
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2. A collection of coins contains nickels, dimes, and quarters. There are 5 more nickels than
dimes, and twice as many quarters as dimes. The value of the collection is $2.20. Find
the number of each coin in the collection.
EQ3 – SP17
Solving Equations 3
3.4 Vocabulary, Skill Builders, and Review
FOCUS ON VOCABULARY
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Use the word bank and vocabulary in this packet to complete the puzzle.
Across
Down
3
1 Twenty-five cents
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5
Quadrilateral where opposite sides are
parallel
Triangle with two equal sides
8
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Statement that asserts two expressions
are true
Prove
11 Picture that organizes information
2 Numbers that include fractions and
integers
4 Ten cents
6 The “-5” in -5x
7 Value that makes an equation true
12 Five cents
EQ3 – SP18
Solving Equations 3
SKILL BUILDER 1
-( x − y )2
-20
x+y
3.
5.
(-x − y )2
2
6. -x + y
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4.
-20 ÷ x + y
2.
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1. -x - (-y)
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Evaluate each expression for x = -10 and y = -20.
-50 + (-60)
9.
40 – 140
8.
200 – (-200)
10.
-140 – 40
12.
-2(20 – 60)
14.
-
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7.
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Compute mentally.
-10 – 10 – (-10)
13.
(-5•20) + (-4•25)
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11.
1 -8
•
2 4
Sa
Solve each equation using a mental strategy.
4.
-8 =
24
y
5.
1
(12 + z ) = 9
2
EQ3 – SP19
Solving Equations 3
•
•
Use the variables below to write expressions that match each purchase.
Then, evaluate each expression.
Jewelry Sale
Jewelry:
$15.00
$12.75
$9.50
$30.00
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Necklace…………………………….......................................
Bracelet ………………………………………………………....
Earrings (pair) ……………………………………………………
Ring…..………………………………......................................
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SKILL BUILDER 2
Let n = the cost of a necklace
Let e = the cost of earrings
Let b = the cost of a bracelet
Let r = the cost of a ring
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1. Purchase 2 bracelets and 3 rings.
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2. Purchase 2 pairs of earrings, one ring, and one bracelet.
Find the value of x to make the equation true.
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1.25 + x = 2.25 + 1.50
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1.
2. 3 x = 3.75
Solve each equation. Draw pictures if needed.
-5( x + 1) = 25
7.
3 x = -3 x − 6
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6.
EQ3 – SP20
Solving Equations 3
SKILL BUILDER 3
Guess
Second even
integer
Third even
integer
Check
Sum
Sum = ______?
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First even
Integer (x)
and
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1. The sum of three consecutive even numbers is 372. What are the three numbers?
x
and
Solution
What is each number?
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Equation
Write an equation and solve it if you can.
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Verify the solution using the equation.
Make and sketch
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2. The length of a rectangle is 5 less than
twice its width. The perimeter is 14 feet.
Find the length and width.
Width
Guess and Check
Length
Perimeter
P = ______?
W
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Write an equation and solve it.
Equation and Solution
How long is each side?
Verify the solution using the equation
EQ3 – SP21
Solving Equations 3
SKILL BUILDER 4
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Chaz went running at the park. Use the graph to complete the table.
0
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300 yd
1 min
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Distance
600 yd
4 min Graph is not drawn to scale.
Time
Distance traveled
1. From 0 minutes to 1 minute
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2. From 1 minute to 4 minutes
Average rate of speed
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Time period
3. From 0 minutes to 4 minutes
m
5. Suppose you poured water into this
container at a constant rate. Sketch a
graph and write one or two sentences to
justify it.
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Explain:
Graph:
Height of Water
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4. In what part of the jog did Chaz run faster, the initial one minute or the last three
minutes? Explain by referencing numbers and the shape of the graph.
Number of Pours
EQ3 – SP22
Solving Equations 3
SKILL BUILDER 5
POOCH CLEANERS
4 washes for $____
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DOGGIE WASHERS
5 washes for $____
cost
(y)
# of
washes
(x)
cost
(y)
4
$32
$25
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# of
washes
(x)
PC
6
$48
$0
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2. Which is the better buy? Use entries in the
tables or graphs to explain your
reasoning.
3. Write equations to relate the number of
dog washes to cost.
y = _____________
PC
y = _____________
m
DW
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4. How can you determine unit rate from the
equation?
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DW
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1. Complete the tables and graphs. Assume a proportional relationship. The graph for
Doggie Washers (DW) is provided. A partial table for Pooch Cleaners (PC) is provided.
Use tables and graphs to complete pricing circles above.
0
5
5. Identify the coordinates when x = 1
DW
(1, ____)
PC
(1, ____)
6. What does this represent in the context of
the problem?
7. Identify the coordinates when x = 0
DW
(0, ____)
PC
(0, ____)
What does this represent in the context of
the problem?
EQ3 – SP23
Solving Equations 3
SKILL BUILDER 6
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Which train is it, and how many hours
have passed?
Guess
Make a sketch (if needed)
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1. Two trains start traveling in the same
direction, from the same place and at the
same time. Train A travels at a rate of 90
miles per hour and Train B travels at a
rate of 120 miles per hour. After some
time has passed, one train is ahead by
150 miles.
Total distance
traveled
____=____?
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Number of hours Distance A Train Distance B Train
(x)
travels
travels
Check
x
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Equation
Write the equation and solve
it.
and
How long will it take for one
train to be ahead by 150
miles?
Solution
Check your answer using the
equation.
Which train is it?
How far does each train
travel?
A→
B→
EQ3 – SP24
Solving Equations 3
SKILL BUILDER 7
Equation/Steps
-5 + 10 x = 15 x +10
What was done?
given equation (nothing done)
-5 + 10 x = 15 x +10
- 10 x - 10 x
subtract 10x from both sides;
addition (subtraction) property of equality
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1.
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Check each solution. If it is correct, write in what was done for each step. If it is incorrect, find
the mistake and correct it. Use pictures if needed.
- 5 = 5 x + 10
- 10
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-10
-1 = x
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Check solution by substitution:
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-5 5 x
=
5
5
Equation/Steps
What was done?
-6( x − 5) = 4 x + 20
given equation (nothing done)
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2.
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-6 x − 5 = 4 x + 20
+6 x
+ 6x
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-5 = 10 x + 20
-20
- 20
-25
10 x
=
10
10
-5
= x
2
EQ3 – SP25
Solving Equations 3
SKILL BUILDER 8
Equation/Steps
What did you do?
(include the property used)
x − 2 x − 5 = 3( x + 5)
given equation
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1.
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Write all the steps used to solve the equations. Provide justifications/explanations for each
step. Use pictures as needed.
Equation/Steps
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2.
What did you do?
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− 2x = - 6 − x + 3
EQ3 – SP26
Solving Equations 3
SKILL BUILDER 9
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For each Problem, underline what you are trying to find. Then answer questions a-d.
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pictures.
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pictures.
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1. The length of the second side of a triangle 2. The sum of three consecutive odd
is 2 less than 3 times the length of the
integers is -111. What are the three
first side. The third side is 4 times longer
numbers?
than the first side. The perimeter of the
triangle is 10 units. How long is each
side?
m
g. Check the solution.
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g. Check the solution.
EQ3 – SP27
Solving Equations 3
SKILL BUILDER 10
3( x + 2) = 36
2.
3.
-3 x − 15 = 2 x
4.
5.
-3(2 x + 1) = -5 x − 4
2( x − 6) = x − 3
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1.
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Solve for x.
6.
-20 x − 8 = -12 x + 24
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- x + 5 = -2( x − 6)
EQ3 – SP28
Solving Equations 3
SKILL BUILDER 11
1
1
= 10 x −
6
5
3.
2
1
x−4 = x+8
3
6
2.
2(x – 4.5) = -2.5 + x
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1.
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Solve each equation.
10.4 – 1.5x = 3.6 x – 30.4
-0.5 x + 4 = 2 x − 6
6.
1

6  x − 1 + x = 3
3

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5.
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4.
EQ3 – SP29
Solving Equations 3
SKILL BUILDER 12
Solve for t:
6t = D
2.
Solve for t:
x – 2y = 7
4.
Solve for y:
rt = D
x – 2y = 7
Solve for H:
LWH = V
6.
Solve for B:
8.
Solve for y:
A=
1
Bh
2
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5.
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3. Solve for x:
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1.
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Solve for the variable indicated.
3x + y = 12
3x + y = 12
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7. Solve for x:
EQ3 – SP30
Solving Equations 3
SKILL BUILDER 13
2x + 18
od
x + 50
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For each problem below, the diagram shows two parallel lines cut by a transversal. Find the
value of x and the value of all the angles in the diagram.
1.
2.
2x - 40
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3x - 50
3.
4.
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2x + 5
2x + 18
145
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100
EQ3 – SP31
Solving Equations 3
Show your work on a separate sheet of paper and choose the best answer.
2. Solve for x:
A.
-24
B.
0
C.
2.5 x + 5 = 2( x − 3.5)
B.
-8
C.
1
D.
2
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-1
8
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A.
1
1
( x − 1) =
2
2
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1. What is the solution to the equation
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TEST PREPARATION
D.
24
14 + n = 72
C.
14 − n = 72
B.
72n = 14
D.
72 + n = 14
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A.
N
3. The sum of a number and 14 is 72. Which equation shows this relationship?
4. The sum of two numbers is 88. One number is 7 times more than the other. What is the
smaller number?
A.
10
B.
11
C.
70
D.
77
Sa
m
5. You have a collection of 15 coins, all dimes and quarters. The total value of the coins is
$2.25. How many dimes do you have?
A.
10
B.
5
C.
-5
D.
-10
EQ3 – SP32
Solving Equations 3
KNOWLEDGE CHECK
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e
Show your work on a separate sheet of paper and write your answers on this page.
3.1 Solving Equations with Rational Coefficients
2
5
= 2x +
3
6
2.
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3.2 Perimeter and Number Problems Revisited
2.5 x + 7.5 = 0.5 x + 16.5
ep
r
x+
1.
od
Solve each equation and justify each step.
Solve.
o
N
3. Three times a number added to 5 times the same number is equal to 176. What is the
number?
pl
e:
D
4. The perimeter of a rectangle is 98 units. The length is 9 units more than the width. Find the
dimensions of the rectangle.
3.3 Coin Problems
m
Solve.
Sa
5. A collection of 13 coins consists of all nickels and dimes. The total value of the coins is
EQ3 – SP33
Solving Equations 3
Sa
m
pl
e:
D
o
N
ot
R
ep
r
od
uc
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This page is left intentionally blank for notes
EQ3 – SP34
Solving Equations 3
Sa
m
pl
e:
D
o
N
ot
R
ep
r
od
uc
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This page is left intentionally blank for notes.
EQ3 – SP35
Solving Equations 3
Sa
m
pl
e:
D
o
N
ot
R
ep
r
od
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This page is left intentionally blank for notes.
EQ3 – SP36
Solving Equations 3
HOME-SCHOOL CONNECTION
5
1
x + x + 11 = -1
2
2
od
1. Solve the equation
uc
e
Here are some questions to review with your young mathematician.
ot
R
ep
r
2. The length of the second side of a triangle is five units less than the length of the first
side. The length of the third side is twice the length of the first side. The perimeter of the
triangle is 63 units. How long is each side?
o
N
3. A collection of 20 coins contains all dimes and quarters. The total value of the coins is
Sa
m
pl
e:
D
Parent (or Guardian) signature ________________________________________________
EQ3 – SP37
Solving Equations 3
COMMON CORE STATE STANDARDS – MATHEMATICS
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SELECTED COMMON CORE STATE STANDARDS FOR MATHEMATICS
Apply properties of operations as strategies to add, subtract, factor, and expand linear
expressions with rational coefficients.
7.EE.4a
Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and
r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic
solution to an arithmetic solution, identifying the sequence of the operations used in each
approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its
width?
8.EE.7b
Solve linear equations with rational number coefficients, including equations whose solutions
require expanding expressions using the distributive property and collecting like terms.
FIRST PRINTING
ep
r
od
7.EE.1
DO NOT DUPLICATE © 2011
STANDARDS FOR MATHEMATICAL PRACTICE
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
MP3
MP4
MP5
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
MP6
MP7
MP8
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Sa
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pl
e:
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N
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MP1
MP2
EQ3 – SP38

Reproduce EQ3 EQUATIONS AND INEQUALITIES STUDENT PACKET 3: SOLVING EQUATION 3

Transcription

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