Download Chapter 3 Study Guide

Chapter 3 Study Guide
By Juan Quintero
Lesson 5
Solving equations with variables on each side
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Focus
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Vocabulary
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Solve equations with the variable on each side.
Solve equations using grouping symbols.
Identity - An equation that is true for every value of the equation*.
* Example of identity on slide 5
Solve -2 + 10p = 8p – 1
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-2 + 10p = 8p – 1
-2 + 10p +2 = 8p – 1 + 2
10p = 8p + 1
10p – 8p = 8p + 1 – 8p
2p = 1
2p = 1
2
2
p=½
add 2 on both sides
subtract 8p from both sides
divide each side by 2
Lesson 5 (cont.)
Solve 6v – 9 = v
3
• 6v – 9 = v
3
• (3) 6v – 9 = (3)v
times each side by 3
3
• 6v – 9 = 3v
That gives you this
• 6v – 9 – 6v = 3v – 6v
Then subtract each side by 6v
• -9 = -3v
Then divide each side by -3
-3
-3
• 3=v
That gives you this as an answer.
Lesson 5 (cont.)
 Solve 6(y – 5) = 18 – 2y
6(y – 5) = 18 – 2y
Use the distributive propery
6y – 30 = 18 – 2y
That gives you this.
6y – 30 + 30 = 18 – 2y + 30 Add 30 to each side.
6y = 48 – 2y
That gives you this
6y + 2y = 48 – 2y + 2y
Then add each side by 2y
8y = 48
Last divide each side by 8
8
8
 y=6
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Lesson 5 (cont.)
Solve 3(r + 1) – 5 = 3r – 2
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3(r + 1) – 5 = 3r – 2
3r + 3 – 5 = 3r – 2
3r – 2 = 3r – 2
3r – 2 – 3r = 3r – 2 – 3r
-2 = -2
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It’s an IDEntItY
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Use the distributive property
Next subtract 5 from 3.
Then subtract each side by 3r.
This Is your answer but they’re the same so…
Solve 2m + 5 = 5(m – 7) – 3m
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2m + 5 = 5(m – 7) – 3m
2m + 5 = 5m – 35 – 3m
2m + 5 = 2m – 35
2m + 5 – 5 = 2m – 35 – 5
2m = 2m – 37
2m – 2m = -37
0 = -37
NO SOLUTION
Use the distributive property
You get this but then subtract 3m from each side.
That gives you this.
Next subtract each side by 5
Last subtract 2m from 2m
But 0 is not equal to -37 so it has no solution.
Lesson 6
Ratios and proportions
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Focus
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Determine whether two ratios form a proportion.
Solve proportions.
Vocabulary
Ratio – is a comparison of two numbers by division.
Proportion – An equation stating that two ratios are equal.
Extremes and means – In the proportion 2 = 14, 21 and 2 are the extremes and 3 and 14
are the means.
3 21
 Rate – The ratio of two measurements having different units of measure.
 Scale – A ratio or rate being used when making a model or drawing of something that is too big to be drawn
at actual size.
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See if these form a porpotion
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5 = 10
10 20
Multiply using cross products
5(20) = 10(10)
Form it to an equation. Then multiply
100 = 100
Yes it is a proportion because each side is equal.
REMEMBER: IN ORDER FOR THE RATIOS TO FORM A PORPORTION THE CROSS PRODUCTS HAVE
TO BE EQUAL.
Lesson 6 (cont.)
Solve the proportion.
•
6=x
5 15
6(15) = 5x
90 = 5x
90 = 5x
5
5
18 = x
multiply using cross product
to form an equation.
That gives you this which is a two step equation
Divide each side by 5
Solve the proportion
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5m – 3 = 5m + 3
4
6
6(5m – 3 ) = 4(5m + 3)
30m – 18 = 20m + 12
30m – 18 + 18 = 20m + 12 + 18
30m = 20m + 30
30m – 20m = 20m – 20m + 30
10m = 30
10 10
m=3
Use cross poducts to form an equation
This is now an equation with variables on both sides.
To get this use the distributive property.
Then add each side by 18
That gives you this
Next subtract each side by 20m.
After that divide each side by 10.
Lesson 6 (cont.)
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Mr. Deranged Guy has driven 315 miles in 5 hours. How much will it take him to drive 255 miles?
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First step make an proportion
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Let h represent the time he can drive 255 mile.
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This is how it should be set up.
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Now solve
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315 = 225 let this row represent the miles.
5
h
let this row be the hours or time.
315 = 225
Set up an equation using cross products
5
h
315(h) = 5(255)
Now multiply
315h = 1275
Then divide each side by 315.
315
315
h = 4.047619 hours
h = 4.04 hours or {4 hours, 2 minutes and 24 seconds (if you do the math)}.
It will take Mr. Deranged Guy 4 hours, 2 minutes and 24 seconds to drive 255 miles.
Lesson 7
Percent of Change
Focus
Find percents of decrease and increase.
Solve problems involving percent of change.
Vocabulary
Percent of change – when a percent of decrease or increase is expressed as an percent.
Percent of increase – the new number is higher than the original number.
Percent of decrease – the new number is lower than the original.
Find Percent of Change.
Original: 45 New: 80
Dfference of the two numbers 35 = r
original number goes here
45 100
35 = r
45 100
Set up an equation using X products.
45(r) = 35(100)
Multiply
45r = 350
Then divide each side by 45
45
45
r = 77.77
r = 78 (rounded)
There’s a 78% increase.
Lesson 7 (cont.)
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Find Percent of Change.
– Original: 125 New: 80
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Difference 45 = r
original # 125 100
45 = r
125 100
45(100) = 125(r)
4500 = 125r
125
125
36 = r
36% percent decrease
Set up an equation
Multiply
Divide each side by 125
Lesson 7 (cont.)
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Mr. Bobeth’s sword is 35 inches long. Mr. Medieval Deranged Guy’s sword is
40% longer. What is the length of Mr. Medieval Deranged Guy’s sword?
Since the 40% is a percent of increase, Mr. Bobeth’s sword is shorter than Mr.
Medieval Deranged Guy’s sword. Let S – 35 be the percent of change.
 Let this represent the amount of change S - 35 = 40
This is the original amount
35
100
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S – 35 = 40
35
100
100(S – 35) = 35(40)
Multiply
100S – 3500 = 1400
100S – 3500 + 3500 = 1400 + 3500
Add 350 to each side.
100S = 4900
Then divide by 100 on both sides.
100
100
S = 49
Mr. Medieval Deranged Guy’s sword is 49 inches.
Lesson 7 (cont.)
0011 0010 1010 1101 0001 0100 1011
• Find amount after Discount.
– Deranged Airlines is having an airfare sale. Mr. Bob reserved a ticket for
$337 to go to Mexico. With the sale he will be saving 45%. What is the
new ticket price?
– $337(45%) = $337(.45) REMEMBER TO MOVE THE DECIMAL 2 PLACES
– $337(45%) = $151.65 MULTIPLY
1
• $337 – $151 = $186 $SUBTRACT $151.65 FROM $337 DOLLARS.
• Bob’s ticket is now $186.
2
4
Lesson 7 (cont.)
• Find amount after tax
– Mr. Deranged Guy bought a new plasma HDTV for $2,999.
The tax is 6.25%. What is the total price of the TV?
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$2,999(6.25%) = $2,999(.0625)
$2,999(6.25%) = $187.4375 or $187.44(rounded).
$2,999 + 187.44 = $3,186.44
The total price for the plasma HDTV is $3,186.44
Lesson 7 (cont.)
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Find the final price.
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Mr. Deranged Guy bought a brand new guitar for
$1,400. The tax is 6% and it has a discount for
15%. Find the final price.
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$1,400(15%) = $1,400(.15) Multuply
$1,400(15%) = $210
That gives you $210
$1,400 – $210 = $1190
Subtract $210 from
$1,400. That gives you the discount.
$1190(6%) = 1190(.06) Next multiply disc., by
tax.
$1190(6%) = $71.40
$1190 + $71.40 = $1261.40
The final price for the guitar is $1261.40.
REMEMBER TO CALCULATE DISCOUNT FIRST THE DO THE TAX
MOVE THE DECIMAL TWO PLACES TO THE LEFT WHEN CHANGING A
PERCENT TO A DECIMAL NUMBER
Lesson 8
How to solve an Equation for a specific variable
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Focus
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Vocabulary
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Solve A= bh for b
– A= bh
– A=bh
First divide h from both sides
h h
– A=b
h
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Solve equations for a specific variable.
Use formulas to solve real world problems.
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Dimensional analysis – is the process of carrying units throughout a computation.
Lesson 8 (cont.)
Solve Q = c+d for d
2
– Q = c+d
2
– (2)Q = 2 .
1
c+d Times each side by 2
2
– 2Q – c = c+d-c
– 2Q – c = d
Subtract c on both sides.
Solve Q = 3a + 5ac for a
– Q = 3a + 5ac
– Q = a(3+5c)
– Q = a(3+5c)
–
3+5c
Q
3+5c
3+5c
=a
distribute the a to 3+5c
divide 3+5c both sides
Lesson 8 (cont.)
 Solve P= 2l + 2w for w
 P = 2l + 2w
 P – 2l = 2l + 2w – 2l
 P – 2l = 2w
2
 P – 2l = w
2
Subtract 2l from both sides
Then divide 2 from both sides
Lesson 8
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Mr. Architect Deranged Guy was measuring a triangle wall. Its base is 134 ft and its area is 4445 ft
sq. He had the measure of the height but he accidentally spilt soda on the measurement and now
he can’t see the measure of the height.
This is the equation on how you find area.
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A = ½ (bh)
Solve for h to find for the height.
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A = ½ (bh)
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A(2) = [½(2)](bh)
2A = bh
2A = bh
b
b
2A = h
b
Plug in the numbers
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2(4445) = h
134
8890 = h
134
66.3432 = h
Since the h is halved in the 1st equation the answer is originally 134.6864 feet or 134.69 feet (rounded).
Mr. Architect Deranged Guy missing measurement is 134.69 feet.
Lesson 9
Weighted averages
• Focus
• Solve mixture problems.
• Solve uniform motion problems.
• Vocabulary
• Weighted averages - the sum of the product of
the number of units and value per unit divided by
the sum of the number units.
• Mixture problems - problems which two or more
parts are combined into a whole.
Lesson 9 (cont.)
•
Mr. Merchant Deranged Guy is selling dried fruit for $4.50 a pound. How
many pounds of mixed nuts selling for $6.05 a pound mixed with 9
pounds of dried fruit to obtain a trail mix that sells for $5.25 a pound?.
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Units
(lbs.)
Set up an equation.
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9(4.50) + 6.05(w) = 5.25(9 + w)
40.50 + 6.05w = 47.25 + 5.25w
40.50 + 6.05w – 6.05w = 47.25 + 5.25w – 6.05w
40.25 = 47.25 – 0.8w
40.25 – 47.25 = 47.25 – 0.8w – 47.25
-7 = -0.8w
-0.8 -0.8
8.75 = w
Mr. Merchant Deranged Guy will put 8.75 pounds of mixed nuts with the
9 pounds of dried fruit.
Price
per
unit
(lbs.)
Total
Price
Dried
fruits
9
$4.50
9(4.50)
Mixed
Nuts
w
$6.05
6.05w
9+w
$5.25
5.25(9 + w)
Trail
Mix
Lesson 9 (cont.)
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Mr. Deranged the Genius is doing an expirament that calls for
a 40% solution of copper sulfate. He has 50 mLof 25%
solution. How many mL of 60% solution should she add to
obtain the required 40% solution?
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50(.25) + .60x = .40(50 + x)
12.5 + .60x = 20 + .40x
12.5 + .60x – 12.5 = 20 + .40x – 12.5
.60x = 7.5 + .40x
.60x - .40x + 7.5 = .40x - .40x
.20x + 7.5 = 0
.20x + 7.5 – 7.5 = 0 – 7.5
.20x = -7.5
.20 .20
X = 37.5 mL
He needs 37.5 mL of the 60% solution.
Amount
25% solution
50
50(.25)
60% solution
x
.60x
40% solution
50 + x
.40(50 + x)
Lesson 9 (cont.)
• Bob’s car and an ambulance are
headed towards each other. The car is
approaching at a speed of 30 mph or
70 ft.ps. The ambulance is
approaching at 50 mph or 74 ft.sp. If
they are 4,000 feet apart then how
may seconds will Bob pass by the
ambulance?
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70t + 74t = 4000
144t = 4000
144t = 4000
144
144
t = 27.77
They’ll pass by in 27.77 seconds.
r
t
d = rt
Bob
70
t
44t
Ambulance
74
t
74t
Lesson 9 (cont.)
•
Bob the businessman has to go to a businessman convention 45 miles
away. He’s in his car traveling with heavy traffic and it took him an hour
to get there. When the event was over he went back to his home with
heavier traffic taking him 3 hours to get home. What is the average speed
he drove throughout the journey?
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45/1 = average 45 mph GOING
45/3 = average 22.5 mph RETURNING
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Average speeds with hours 45(1) + 15(3)
The hours
3+1
45 + 45
3+1
90
4
22.5 The average speed was 22.5 mph.
THE END
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BY JUAN