Download Rewriting Equations and Formulas

Topic 3 Rewriting Equations and Formulas
Lesson Tutorials
An equation that has two or more variables is called a literal equation.
To rewrite a literal equation, solve for one variable in terms of the other
variable(s).
Key Vocabulary
literal equation,
p. 774
EXAMPLE
1
Rewriting an Equation
Solve the equation 2y + 5x = 6 for y.
2y + 5x = 6
Undo the addition.
Write the equation.
2y + 5x − 5x = 6 − 5x
Subtraction Property of Equality
2y = 6 − 5x
2y
2
Simplify.
6 − 5x
2
—=—
Undo the multiplication.
Division Property of Equality
5
2
y = 3 − —x
Simplify.
Solve the equation for y.
1. 5y − x = 10
Exercises 5 –10
EXAMPLE
Remember
A formula shows how
one variable is related
to one or more other
variables. A formula is a
type of literal equation.
2
2.
4x − 4y = 1
3.
12 = 6x + 3y
Rewriting a Formula
The formula for the perimeter P of a rectangle is P = 2ℓ + 2w.
Solve the formula for the length ℓ.
P = 2ℓ + 2w
Write the formula.
P − 2w = 2ℓ + 2w − 2w
Subtraction Property of Equality
P − 2w = 2ℓ
Simplify.
2ℓ
P − 2w
—=—
Division Property of Equality
P − 2w
2
Simplify.
2
2
— =ℓ
w
Solve the formula for the red variable.
Exercises 14 – 19
4. Area of rectangle: A = bh
5.
Simple interest: I = Prt
6. Height of a dropped object: h = −16t 2 + s
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Additional Topics
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COMMON
CORE
Solving Equations
In this lesson, you will
● rewrite equations to solve
for one variable in terms
of the other variable(s).
Applying Standard
8.EE.7
EXAMPLE
Temperature Conversion
A formula for converting from degrees Fahrenheit F to degrees
Celsius C is
5
9
C = — (F − 32).
Rewriting the Temperature Formula
3
Solve the temperature formula for F.
5
9
C = — (F − 32)
9
5
—
Use the reciprocal.
Write the temperature formula.
⋅ C = —95 ⋅ —59 (F − 32)
Multiplication Property of Equality
9
5
Simplify.
— C = F − 32
Undo the subtraction.
— C + 32 = F − 32 + 32
9
5
Addition Property of Equality
9
5
Simplify.
— C + 32 = F
9
5
The rewritten formula is F = — C + 32.
EXAMPLE
4
Real-Life Application
Which has the greater temperature?
Sun
11,000°F
Convert the Celsius temperature of lightning to Fahrenheit.
9
5
Write the rewritten formula from Example 3.
= — (30,000) + 32
9
5
Substitute 30,000 for C.
= 54,032
Simplify.
F = — C + 32
Lightning
30,000°C
Because 54,032 °F is greater than 11,000 °F, lightning has the
greater temperature.
7. Room temperature is considered to be 70 °F. Suppose
the temperature is 23 °C. Is this greater than or less than
room temperature?
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Topic 3 Exercises
Help with Homework
3
8
1. VOCABULARY Is −2x = — a literal equation? Explain.
2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.
Solve 4x − 2y = 6 for y.
Solve 6 = 4x − 2y for y.
Solve 4x − 2y = 6 for y in terms of x.
Solve 4x − 2y = 6 for x in terms of y.
6)=3
9+(- 3)=
3+(- 9)=
4+(- =
1)
9+(-
3. a. Write a formula for the area A
of a triangle.
4.
a. Write a formula for the volume V
of a prism.
b. Solve the formula for b.
b. Solve the formula for B.
c. Use the new formula to find the
base of the triangle.
c.
Use the new formula to find the
area of the base of the prism.
V â 36 in.3
A â 36 mm2
h â 6 in.
h â 6 mm
B
b
Solve the equation for y.
1
1
3
1
5
5. — x + y = 4
6. 3x + — y = 7
8. 3 = 7x − 2y
9. 4.2x − 1.4y = 2.1
11. ERROR ANALYSIS Describe and correct the
error in rewriting the equation.
7. 6 = 4x + 9y
✗
10. 6y − 1.5x = 8
2x − y = 5
y = −2x + 5
12. TEMPERATURE The formula K = C + 273.15
converts temperatures from Celsius C to Kelvin K.
a. Solve the formula for C.
b. Convert 300 Kelvin to Celsius.
I
$75
P
$500
a. Solve the formula for t.
r
5%
b. Use the new formula to find the value of t in the table.
t
13. INTEREST The formula for simple interest is I = Prt.
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Solve the equation for the red variable.
15. e = mc 2
2 14. d = rt
1
2
16. R − C = P
V
h
17. A = — h(b1 + b2)
1
6
18. B = 3 —
19. g = —(w + 40)
20. LOGIC Why is it useful to rewrite a formula in terms
of another variable?
5
9
21. REASONING The formula K = — (F − 32) + 273.15
converts temperatures from Fahrenheit F to Kelvin K.
a. Solve the formula for F.
b. The freezing point of water is 273.15 Kelvin.
What is this temperature in Fahrenheit?
c. The temperature of dry ice is −78.5 °C.
Which is colder, dry ice or liquid nitrogen?
Navy Pier Ferris Wheel
C = 439.6 ft
77.35 K
22. FERRIS WHEEL The distance around a circle is called
the circumference. The Navy Pier Ferris Wheel in
Chicago has a circumference C that is 56% of the
circumference of the first Ferris wheel built in 1893.
a. The radius of a circle is the distance from the center
to any point on the circle. The circumference of a
circle is about 6.28 times the radius. What is the
radius of the Navy Pier Ferris Wheel?
b. What was the radius of the first Ferris wheel?
c. The first Ferris wheel took 9 minutes to make a
complete revolution. How fast was the wheel moving?
23.
Repeated
Reasoning The radius of a sphere is the distance from
the center to any point on the sphere. The formula for the
4
3
volume V of a sphere is V = —π r 3, where r is the radius and π
is a constant whose value is about 3.14. Solve the formula for r 3.
Use Guess, Check, and Revise to find the radius of the sphere.
V â 121.5Ĭ in.3
r
Multiply. (Skills Review Handbook)
3
4
24. 5 × —
8
3
25. −2 × —
1
4
3
2
8
9
3
5
26. — × — × —
1
12
27. 25 × — × —
3
28. MULTIPLE CHOICE Which of the following is not equivalent to — ?
4
(Skills Review Handbook)
A 0.75
○
B 3:4
○
C 75%
○
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○
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