Download 1.4Rewrite Formulas and Equations

1.4
Before
Now
Why?
Key Vocabulary
• formula
• solve for a variable
Rewrite Formulas
and Equations
You solved equations.
You will rewrite and evaluate formulas and equations.
So you can apply geometric formulas, as in Ex. 36.
A formula is an equation that relates two or more quantities, usually represented
by variables. Some common formulas are shown below.
Quantity
READING
The variables b1 and b2
are read as “b sub one”
and “b sub two.” The
small lowered numbers
are called subscripts.
Formula
Distance
d 5 rt
Temperature
F 5 } C 1 32
Area of a triangle
Meaning of variables
d 5 distance, r 5 rate, t 5 time
9
5
F 5 degrees Fahrenheit,
C 5 degrees Celsius
A 5 } bh
1
2
A 5 area, b 5 base, h 5 height
Area of a rectangle
A 5 lw
A 5 area, l 5 length, w 5 width
Perimeter of a rectangle
P 5 2l 1 2w
Area of a trapezoid
A 5 } (b1 1 b2)h
A 5 area, b1 5 one base,
b2 5 other base, h 5 height
Area of a circle
A 5 πr2
A 5 area, r 5 radius
Circumference of a circle
C 5 2π r
C 5 circumference, r 5 radius
1
2
P 5 perimeter,
l 5 length, w 5 width
To solve for a variable means to rewrite an equation as an equivalent equation in
which the variable is on one side and does not appear on the other side.
EXAMPLE 1
Rewrite a formula with two variables
Solve the formula C 5 2p r for r. Then find the radius of a circle with a
circumference of 44 inches.
Solution
STEP 1
Solve the formula for r.
C 5 2πr
Write circumference formula.
C
2π
}5r
Divide each side by 2p.
STEP 2 Substitute the given value into the rewritten formula.
C
44 ø 7
r5}
5}
2π
2π
Substitute 44 for C and simplify.
c The radius of the circle is about 7 inches.
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Chapter 1 Equations and Inequalities
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✓
GUIDED PRACTICE
for Example 1
1. Find the radius of a circle with a circumference of 25 feet.
2. The formula for the distance d between opposite vertices
2a where a is the distance
of a regular hexagon is d 5 }
}
Ï3
d
a
between opposite sides. Solve the formula for a. Then find
a when d 5 10 centimeters.
EXAMPLE 2
Rewrite a formula with three variables
Solve the formula P 5 2l 1 2w for w. Then find the width
of a rectangle with a length of 12 meters and a perimeter
of 41 meters.
P 5 41 m
12 m
Solution
STEP 1
w
Solve the formula for w.
P 5 2l 1 2w
Write perimeter formula.
P 2 2l 5 2w
Subtract 2l from each side.
P 2 2l
2
Divide each side by 2.
}5w
STEP 2 Substitute the given values into the rewritten formula.
41 2 2(12)
2
w5}
Substitute 41 for P and 12 for l.
w 5 8.5
Simplify.
c The width of the rectangle is 8.5 meters.
"MHFCSB
✓
GUIDED PRACTICE
at classzone.com
for Example 2
3. Solve the formula P 5 2l 1 2w for l. Then find the length of a rectangle with
a width of 7 inches and a perimeter of 30 inches.
4. Solve the formula A 5 lw for w. Then find the width of a rectangle with a
length of 16 meters and an area of 40 square meters.
Solve the formula for the variable in red. Then use the given information to
find the value of the variable.
1 bh
5. A 5 }
2
1 bh
6. A 5 }
2
1 (b 1 b )h
7. A 5 }
2
2 1
b1
h
h
h
b2
b
b
Find h if b 5 12 m
and A 5 84 m 2.
Find b if h 5 3 cm
and A 5 9 cm 2.
Find h if b1 5 6 in.,
b2 5 8 in., and A 5 70 in.2
1.4 Rewrite Formulas and Equations
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REWRITING EQUATIONS The approach you use to solve a formula for a variable
can be applied to other algebraic equations.
EXAMPLE 3
Rewrite a linear equation
Solve 9x 2 4y 5 7 for y. Then find the value of y when x 5 25.
Solution
Solve the equation for y.
STEP 1
9x 2 4y 5 7
Write original equation.
24y 5 7 2 9x
AVOID ERRORS
Subtract 9x from each side.
7 1 9x
y 5 2}
}
4
When dividing each
side of an equation
by the same number,
remember to divide
every term by the
number.
Divide each side by 24.
4
STEP 2 Substitute the given value into the rewritten equation.
7 1 9 (25)
y 5 2}
}
Substitute 25 for x.
7 2 45
y 5 2}
}
Multiply.
y 5 213
Simplify.
4
4
4
4
9x 2 4y 5 7
CHECK
Write original equation.
9(25) 2 4(213) 0 7
Substitute 25 for x and 213 for y.
757✓
EXAMPLE 4
Solution checks.
Rewrite a nonlinear equation
Solve 2y 1 xy 5 6 for y. Then find the value of y when x 5 23.
Solution
AVOID ERRORS
STEP 1
If you rewrite the
equation as
6 2 2y
y5}
x ,
then you have not
solved for y because y
still appears on both
sides of the equation.
Solve the equation for y.
2y 1 xy 5 6
Write original equation.
(2 1 x)y 5 6
Distributive property
6
y5}
21x
STEP 2 Substitute the given value into the rewritten equation.
6
y5}
Substitute 23 for x.
y 5 26
Simplify.
2 1 (23)
✓
Divide each side by (2 1 x).
GUIDED PRACTICE
for Examples 3 and 4
Solve the equation for y. Then find the value of y when x 5 2.
8. y 2 6x 5 7
11. 2x 1 5y 5 21
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9. 5y 2 x 5 13
10. 3x 1 2y 5 12
12. 3 5 2xy 2 x
13. 4y 2 xy 5 28
Chapter 1 Equations and Inequalities
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EXAMPLE 5
Solve a multi-step problem
MOVIE RENTAL A video store rents new movies
for one price and older movies for a lower price,
as shown at the right.
• Write an equation that represents the store’s
monthly revenue.
• Solve the revenue equation for the variable
representing the number of new movies
rented.
• The owner wants $12,000 in revenue per
month. How many new movies must be
rented if the number of older movies rented
is 500? 1000?
Solution
STEP 1
Write a verbal model. Then write an equation.
Monthly
revenue
(dollars)
R
Price of
new movies
5
p
Number of
new movies
(dollars/movie)
5
5
1
(movies)
p
n1
Price of
older movies
p
(dollars/movie)
1
3
Number of
older movies
(movies)
p
n2
An equation is R 5 5n1 1 3n2.
STEP 2 Solve the equation for n1.
R 5 5n1 1 3n2
R 2 3n2 5 5n1
Write equation.
Subtract 3n2 from each side.
R 2 3n2
} 5 n1
5
Divide each side by 5.
STEP 3 Calculate n1 for the given values of R and n2.
12,000 2 3 p 500
5
If n2 5 500, then n1 5 } 5 2100.
12,000 2 3 p 1000
5
If n2 5 1000, then n1 5 } 5 1800.
c If 500 older movies are rented, then 2100 new movies must be rented.
If 1000 older movies are rented, then 1800 new movies must be rented.
✓
GUIDED PRACTICE
for Example 5
14. WHAT IF? In Example 5, how many new movies must be rented if the
number of older movies rented is 1500?
15. WHAT IF? In Example 5, how many new movies must be rented if customers
rent no older movies at all?
16. Solve the equation in Step 1 of Example 5 for n2.
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1.4
EXERCISES
HOMEWORK
KEY
5 WORKED-OUT SOLUTIONS
on p. WS1 for Exs. 3, 9, and 35
★
5 STANDARDIZED TEST PRACTICE
Exs. 2, 6, 15, 27, 36, and 38
SKILL PRACTICE
1. VOCABULARY Copy and complete: A(n) ? is an equation that relates two or
more quantities.
2. ★ WRITING What does it mean to solve for a variable in an equation?
EXAMPLES
1 and 2
on pp. 26–27
for Exs. 3–6
REWRITING FORMULAS Solve the formula for the indicated variable. Then use
the given information to find the value of the variable.
3. Solve A 5 lw for l. Then find the length of a rectangle with a width of
50 millimeters and an area of 250 square millimeters.
1 bh for b. Then find the base of a triangle with a height of
4. Solve A 5 }
2
6 inches and an area of 24 square inches.
1 (b 1 b )h for h. Then find the height of a trapezoid with
5. Solve A 5 }
2
2 1
bases of lengths 10 centimeters and 15 centimeters and an area of
75 square centimeters.
6. ★ MULTIPLE CHOICE What equation do you obtain when you solve the
1 (b 1 b )h for b ?
formula A 5 }
2
1
2 1
2A 2 b
A b1 5 }
2
A 2b
B b1 5 }
2
C b1 5 2A 2 b2h
2A
D b1 5 }
h
2h
h 2 b2
EXAMPLE 3
REWRITING EQUATIONS Solve the equation for y. Then find the value of y for the
on p. 28
for Exs. 7–17
given value of x.
7. 3x 1 y 5 26; x 5 7
8. 4y 1 x 5 24; x 5 8
9. 6x 1 5y 5 31; x 5 24
10. 15x 1 4y 5 9; x 5 23
11. 9x 2 6y 5 63; x 5 5
12. 10x 2 18y 5 84; x 5 6
13. 8y 2 14x 5 222; x 5 5
14. 9y 2 4x 5 230; x 5 8
15. ★ MULTIPLE CHOICE What equation do you obtain when you solve the
equation 4x 2 5y 5 20 for y?
5y 1 5
A x5}
4
4x 1 4
B y 5 2}
5
4x 2 4
C y5}
4 x 2 20
D y5}
5
5
ERROR ANALYSIS Describe and correct the error in solving the equation for y.
16.
17.
27x 1 5y 5 2
5y 5 7x 1 2
4y 5 9 1 xy
7x 1 2
y5}
y5}
5
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4y 2 xy 5 9
9 1 xy
4
Chapter 1 Equations and Inequalities
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GEOMETRY Solve the formula for the variable in red. Then use the given
information to find the value of the variable. Round to the nearest tenth.
18. Area of a
circular ring
19. Lateral surface area
of a truncated cylinder
S 5 πr(h 1 k)
A 5 2πrw
20. Volume of
an ellipsoid
4 πabc
V5}
3
b
w
r
a
c
k
h
r
Find r if w 5 4 ft
and A 5 120 ft 2.
Find h if r 5 2 cm,
k 5 3 cm, and S 5 50 cm 2.
Find c if a 5 4 in.,
b 5 3 in., and V 5 60 in.3
EXAMPLE 4
REWRITING EQUATIONS Solve the equation for y. Then find the value of y for the
on p. 28
for Exs. 21–26
given value of x.
21. xy 2 3x 5 40; x 5 5
22. 7x 2 xy 5 218; x 5 24
23. 3xy 2 28 5 16x; x 5 4
24. 9y 1 6xy 5 30; x 5 26
25. y 2 2xy 5 15; x 5 21
26. 4x 1 7y 1 5xy 5 0; x 5 1
27. ★ SHORT RESPONSE Consider the equation 15x 2 9y 5 27. To find the value
of y when x 5 2, you can use two methods.
Method 1 Solve the original equation for y and then substitute 2 for x.
Method 2 Substitute 2 for x and then solve the resulting equation for y.
Show the steps of the two methods. Which method is more efficient if you
need to find the value of y for several values of x? Explain.
REASONING Solve for the indicated variable.
28. Solve xy 5 x 1 y for y.
29. Solve xyz 5 x 1 y 1 z for z.
1 1 1 5 1 for y.
30. Solve }
}
x
y
1 1 1 1 1 5 1 for z.
31. Solve }
}
}
x
y
z
32. CHALLENGE Write a formula giving the area of a circle in terms of its
circumference.
PROBLEM SOLVING
EXAMPLE 5
on p. 29
for Exs. 33–38
33. TREE DIAMETER You can estimate the diameter of a tree without boring
through it by measuring its circumference. Solve the formula C 5 πd for d.
Then find the diameter of an oak that has a circumference of 113 inches.
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
34. DESIGN The fabric panels on a kite are rhombuses. A formula for the length
}
of the long diagonal d is d 5 sÏ 3 where s is the length of a side. Solve the
formula for s. Then find the value of s when d 5 15 inches.
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
s
s
d
s
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35. TEMPERATURE The formula for converting temperatures from degrees
9 C 1 32. Solve the formula for C.
Celsius to degrees Fahrenheit is F 5 }
5
Then find the temperature in degrees Celsius that corresponds to 508F.
36. ★ EXTENDED RESPONSE A quarter mile running track is
shaped as shown. The formula for the inside perimeter
is P 5 2πr 1 2x.
a. Solve the perimeter formula for r.
r
r
b. For a quarter mile track, P 5 440 yards. Find r when
x 5 75 yards, 100 yards, 120 yards, and 150 yards.
c. What are the greatest and least possible values of r if
x
P 5 440 yards? Explain how you found the values, and
sketch the track corresponding to each extreme value.
37. MULTI-STEP PROBLEM A tuxedo shop rents classic tuxedos for $80 and
designer tuxedos for $150. Write an equation that represents the shop’s
revenue. Solve the equation for the variable representing the number of
designer tuxedos rented. The shop owner wants $60,000 in revenue during
prom season. How many designer tuxedos must be rented if the number of
classic tuxedos rented is 600? 450? 300?
38. ★ OPEN-ENDED MATH The volume of a donut-like shape
called a torus is given by the formula V 5 2π 2r 2 R where r
and R are the radii shown and r ≤ R.
r
R
r
R
a. Solve the formula for R.
b. A metal ring in the shape of a torus has a volume of
w
100 cubic centimeters. Choose three possible values of
r, and find the corresponding values of R.
l
39. CHALLENGE A rectangular piece of paper with length l and width
w can be rolled to form the lateral surface of a cylinder in two
ways, assuming no overlapping. Write a formula for the volume of
each cylinder in terms of l and w.
l
w
MIXED REVIEW
PREVIEW
Write an expression to answer the question. (p. 984)
Prepare for
Lesson 1.5
in Exs. 40–47.
40. You have $250 in a bank account and deposit x dollars. What is your current
balance?
41. You buy x CDs for $12.99 each. How much do you spend?
Evaluate the expression for the given value of the variable. (p. 10)
42. 6j 1 8 when j 5 23
43. 6 1 4k 4 2 when k 5 3
44. 8g 2 8g p 2 when g 5 21
45. 25m3 1 m2 when m 5 10
46. (n 1 7)2 2 4 when n 5 2
47. (3p 2 17) 3 when p 5 5
Solve the equation. Check your solution. (p. 18)
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48. 4x 1 7 5 10x 1 25
49. 15 2 2y 5 2y 2 45
50. 56 5 4(4 1 2z)
51. 9(p 1 3) 5 35p 1 1
5q 2 9 5 1
52. }
3
1r 1 1r 5 5
53. }
}
4
6
Chapter 1 EXTRA
EquationsPRACTICE
and Inequalities
for Lesson 1.4, p. 1010
ONLINE QUIZ at classzone.com
10/19/05 2:58:54 PM
MIXED REVIEW of Problem Solving
STATE TEST PRACTICE
classzone.com
Lessons 1.1–1.4
1. MULTI-STEP PROBLEM There is a $50 annual
membership fee to join an urban car rental
service. Using a car costs $8.50 per hour.
a. Write a verbal model for this situation. Then
use the verbal model to write an algebraic
expression.
b. How much will it cost to join the service and
drive for 20 hours?
2. MULTI-STEP PROBLEM You are attending a
museum. You have $50 to spend. Admission to
the museum is $15. Admission to each special
exhibit inside the museum is $10.
5. GRIDDED ANSWER You drive from Chicago to
St. Louis, a distance of 290 miles. Your average
speed is 60 miles per hour. How many hours
does the trip take? Round your answer to the
nearest tenth of an hour.
6. OPEN-ENDED Describe a shopping situation
that can be modeled by the equation
10x 1 29y 5 78.
7. EXTENDED RESPONSE In one year, the Bureau
of Engraving and Printing printed $10 and $20
bills with a total value of $66,368,000. The total
number of $10 and $20 bills was 3,577,600.
a. Write an equation that can be used to find
the number of special exhibits you can
include in your visit.
b. Solve the equation. Interpret your answer in
terms of the problem.
3. SHORT RESPONSE In hockey, each player has
a statistic called plus/minus, which is the
difference between the number of goals scored
by the player’s team and the number of goals
scored by the other team when the player is on
the ice. List the players shown in order from
least to greatest plus/minus. Whose plus/
minus score is best? Explain.
Player
Number
Value
$10 bills
x
10x
$20 bills
?
?
Total
3,577,600
66,368,000
a. Copy and complete the table.
b. Write and solve an equation to find how
many $10 bills and how many $20 bills were
printed.
c. Compare the total value of the $10 bills
printed with the total value of the $20 bills
printed.
Plus/Minus
Vincent Lecavalier
23
Dave Andreychuk
29
Ruslan Fedotenko
14
Martin St. Louis
35
Cory Sarich
5
Tim Taylor
25
4. SHORT RESPONSE You are in charge of buying
food for a school picnic. You have $45 to spend
on ground beef and chicken. Ground beef costs
$1.80 per pound and chicken costs $1.00 per
pound. Write an equation representing the
situation. You want to buy equal amounts
of ground beef and chicken. How much of
each can you buy? Show how you found your
answer.
8. OPEN-ENDED You have two summer jobs. You
mow lawns for $20 per lawn. You also work at a
restaurant for $7.50 per hour. Write an equation
for the total amount of money you earn. Then
find three different ways to earn $300 during
one week.
9. GRIDDED ANSWER The liopleurodon, a
swimming dinosaur from the Late Jurassic
period, could grow to 25 meters in length.
Use the fact that 1 in. 5 2.54 cm to convert
the length to feet. Round your answer to the
nearest foot.
10. GRIDDED ANSWER The formula for the volume
1 Bh. Find h (in centimeters) if
of a cone is V 5 }
3
V 5 176 cm3 and B 5 40 cm 2.
Mixed Review of Problem Solving
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