4. Rewriting and Solving Literal Equations
Sign In
An error ocurred, try again later!
Chapter 1
4.
Rewriting and Solving Literal Equations
This lesson delves into the realm of literal equations, emphasizing their significance in mathematical and real-world scenarios. Literal equations, primarily composed of variables, are pivotal in expressing relationships between different quantities. For instance, the area of a triangle or the volume of a cylinder can be represented using these equations. The lesson also highlights the process of isolating variables, a crucial step in solving these equations. Techniques such as the Properties of Equality and inverse operations play a vital role in this process. Real-life examples, like calculating the time taken for a truck to cover a certain distance or determining the height of a cylindrical object, further elucidate the practical applications of these equations. The lesson not only provides theoretical knowledge but also challenges learners with exercises to test their understanding.
Lesson Settings & Tools
| | 10 Theory slides |
| | 13 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Rewriting and Solving Literal Equations
Slide of 10
When solving real-life problems, it is usually possible to write equations in more than one variable. Sometimes solving the equation for the variable of interest is needed. In this lesson, equations involving two or more variables will be solved for a particular variable.
Catch-Up and Review
Here is some recommended reading before getting started with this lesson.
Share
Report error
Translate
Challenge
Rewriting the Formula for the Perimeter of a Triangle
Consider △ ABC with side lengths a, b, and c.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["P","a","b","c"],"constants":[]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["P=a+b+c","P=a+c+b","P=b+a+c","P=b+c+a","P=c+a+b","P=c+b+a","a+b+c=P"]}}
b If P, a, and b are known, write an equation by solving the formula for c.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["P","a","b","c"],"constants":[]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["c=P-a-b","c= P-(a+b)","c=P-b-a","c= P-(b+a)"]}}
Share
Report error
Translate
Challenge
Finding the Amount of Time Spent Jogging
Maya plans to jog 150 meters at a constant speed of 2.5 meters per second.
The distance traveled d is found using the following equation. d= v* t In this equation, v is the speed and t is the time.
a Determine how long it takes Maya to jog this distance by trying some values for t until a true statement is obtained.
b Find a different way to solve the question.
Share
Report error
Translate
Discussion
Equivalent Equations
Two equations are called equivalent equations if they have the same solution. Equations are often solved by applying the Properties of Equality. Each time a property is applied, an equivalent equation is produced. Consider the following equation. y + 3 = 18 To solve this equation, 3 must be subtracted from both sides.
y + 3 = 18
SubEqn
LHS- 3=RHS- 3
y + 3 - 3= 18 - 3
Simplify left-hand side
y + 3 - 3=18-3
SubTerms
Subtract terms
y=15
Share
Report error
Translate
Example
Determining Equivalent Equations
Dominika is finishing up her homework on Friday so she can enjoy the weekend. She is told that three of the following equations are equivalent equations. Equations [-1em] rl I: & t+11 =20 [0.5em] II: & 7-t = 16 [0.5em] III: & t/3+7 = 10 [0.8em] IV: & 5 = 5t-40 Help Dominika identify the equation that is not equivalent to the other three.
{"type":"choice","form":{"alts":["Equation I","Equation II","Equation III","Equation IV"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}
Hint
Equivalent equations have the same solution(s). Use the Properties of Equality and inverse operations to solve for the variable t.
Solution
Each equation will be solved for t, and then the solutions will be compared.
Equation I
In the first equation, 11 is being added to t. Since the inverse operation of addition is subtraction, applying the Subtraction Property of Equality would simplify the equation.
The solution to the first equation is t=9.
Equation II
To isolate the variable, the Subtraction Property of Equality will be applied first.
7-t =16
SubEqn
LHS-7=RHS-7
7-t - 7=16 - 7
CommutativePropAdd
Commutative Property of Addition
7-7-t=16-7
SubTerms
Subtract terms
- t=9
To remove the negative sign from the variable t, each side of the equation will be divided by - 1. In other words, the Division Property of Equality will be applied.
- t=9
DivEqn
.LHS /(- 1).=.RHS /(- 1).
- t/- 1= 9/- 1
▼
Solve for t
DivNegNeg
- a/- b=a/b
t/1= 9/- 1
MoveNegDenomToFrac
Put minus sign in front of fraction
t/1= - 9/1
DivByOne
a/1=a
t = - 9
The solution to Equation II is t=- 9.
Equation III
To solve Equation III for t, 7 should be subtracted from both sides of the equation.
To remove the fraction, the equation needs to be multiplied by 3. To do so, apply the Multiplication Property of Equality.
t/3=3
MultEqn
LHS * 3=RHS* 3
t/3 * 3=3 * 3
FracMultDenomToNumber
a/3* 3 = a
t=3* 3
Multiply
Multiply
t = 9
The solution to Equation III is t=9, which is the same as the solution to Equation I. Therefore, the first and third equations are equivalent.
Equation IV
In this equation, 40 is being subtracted from 5t. Since the inverse operation of subtraction is addition, applying the Addition Property of Equality will simplify the equation.
By applying the Division Property of Equality, the equation can be further simplified.
45 = 5t
DivEqn
.LHS /5.=.RHS /5.
45/5 = 5t/5
CalcQuot
Calculate quotient
9 = 5t/5
MovePartNumRight
a* b/c=a/c* b
9 = 5/5t
QuotOne
a/a=1
9 = 1t
IdPropMult
Identity Property of Multiplication
9 = t
RearrangeEqn
Rearrange equation
t = 9
The last equation has the same solution as Equations I and III. This means that Equations I, III, and IV are equivalent equations. Equaivalent Equations & Solution I, III, and IV & 9 Therefore, Equation II is not equivalent to any of those equations.
Share
Report error
Translate
Discussion
Literal Equation
A literal equation is an equation that is comprised mostly or entirely of variables. The area of a triangle, for example, is expressed by the following literal equation. A = 1/2bh
Here, b represents the length of the base and h represents the height of the triangle. Therefore, formulas can be seen as literal equations. Using the Properties of Equality, a literal equation can be solved for a variable, which means isolating the variable on one side of the equation.
Share
Report error
Translate
Example
Solving Literal Equations
While at the planetarium on Saturday, Dominika learns about Annie Jump Cannon, known as the census taker of the sky,
who introduced the Harvard Spectral Classification. In its simplest form, this system classifies stars according to their surface temperatures.
| Stellar Classification | ||
|---|---|---|
| Class | Temperature (K) | Apparent Color |
| O | ≥ 30 000 | blue |
| B | 10 000 - 30 000 | blue white |
| A | 7500 - 10 000 | white |
| F | 6000-7500 | yellow white |
| G | 5000 -6000 | yellow |
| K | 3500- 5000 | light orange |
| M | 2000 - 3500 | orange red |
a To convert temperatures in degrees Celsius C to temperatures in Kelvin K, the following literal equation is used.
K = C+273 If Sirius has a surface temperature of 9667^(∘)C, what color does it appear?
b The following literal equation shows the relationship between temperatures in Kelvin K and temperatures in degrees Fahrenheit F.
K= 59(F-32)+273 Solve the formula for F.
Answer
a White
b F=9/5(K-273)+32
c 5840 - 8540
Hint
a Substitute the value into the given equation.
b Use the Properties of Equality and inverse operations to isolate F.
c Substitute the minimum and maximum range values for a light orange star into the derived formula from Part B.
Solution
a To determine the apparent color of Sirius, its surface temperature should be converted into Kelvin. To do so, 9667 will be substituted into the given formula.
The surface temperature of Sirius is about 9940 K. Since its temperature is between 7500 K and 10 000 K, it appears as a white star in the night sky.
b In the given formula, F is the variable of interest. To isolate F, inverse operations will be used.
K=5/9(F-32)+273
SubEqn
LHS-273=RHS-273
K -273=5/9(F-32)
MultEqn
LHS * 9/5=RHS* 9/5
9/5(K -273)=F-32
AddEqn
LHS+32=RHS+32
9/5(K -273) + 32 =F
RearrangeEqn
Rearrange equation
F=9/5(K -273) + 32
This formula can now be used to convert Kelvin to degrees Fahrenheit.
c It is known that Pollux appears as a light orange point of light. Therefore, its surface temperature is between 3500 and 5000 Kelvin. These values should be substituted into the derived formula from Part B. Start with minimum value 3500.
F=9/5(K -273) + 32
Substitute
K= 3500
F=9/5( 3500 -273) + 32
▼
Evaluate right-hand side
SubTerm
Subtract term
F=9/5(3227) + 32
MoveRightFacToNum
a/c* b = a* b/c
F=29 043/5 + 32
CalcQuot
Calculate quotient
F=5808.6+32
AddTerms
Add terms
F=5840.6
RoundInt
Round to nearest integer
F ≈ 5841
Therefore, 3500 Kelvin is approximately equivalent to 5841^(∘) Fahrenheit. Next, the maximum value 5000 will be substituted.
F=9/5(K -273) + 32
Substitute
K= 5000
F=9/5( 5000 -273) + 32
▼
Evaluate right-hand side
SubTerm
Subtract term
F=9/5(4727) + 32
MoveRightFacToNum
a/c* b = a* b/c
F=42 543/5 + 32
CalcQuot
Calculate quotient
F=8508.6+32
AddTerms
Add terms
F=8540.6
RoundInt
Round to nearest integer
F ≈ 8541
This means that 3500 Kelvin is approximately 8541^(∘) Fahrenheit. Therefore, the range for a light orange star in degrees Fahrenheit is as follows. Range(^(∘)F) 5840 - 8540
Share
Report error
Translate
Pop Quiz
Practice Solving Literal Equations
Use the Properties of Equality to solve the given literal equation for the indicated variable. Some equations may require dividing both sides of the equation by a variable. Since division by zero is not defined, for these equations assume that the variable is not equal to 0.
Share
Report error
Translate
Example
Rewriting a Formula for Volume
On Sunday Dominika spends time on her hobby, which is candle making. She uses 170 cubic centimeters of wax to produce a cylindrical candle of radius 3 centimeters.
The volume of a cylinder with radius r and height h is given by the following formula. V= π r^2 h
a Solve the formula for h.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["V","r"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":"h=","formTextAfter":null,"answer":{"text":["\\dfrac{V}{\\pi r^2}"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":""},"formTextBefore":"h \u2248","formTextAfter":"cm","answer":{"text":["6"]}}
Hint
a Use the Division Property of Equality.
b Substitute the given values into the derived formula.
Solution
a On the right-hand side of the formula, h is multiplied by π r^2. V= π r^2 h
To isolate h, the inverse operation of multiplication should be applied to both sides of the equation. This means applying the Division Property of Equality.
V=π r^2h
DivEqn
.LHS /π r^2.=.RHS /π r^2.
V/π r^2=π r^2h/π r^2
CancelCommonFac
Cancel out common factors
V/π r^2=π r^2h/π r^2
SimpQuot
Simplify quotient
V/π r^2=h
RearrangeEqn
Rearrange equation
h = V/π r^2
b It is known that 170 cubic centimeters of wax were used, and that the radius of the cylindrical candle is 3 centimeters. Therefore, V=170 and r=3 will be substituted into the derived formula.
The height of the candle is about 6 centimeters.
Share
Report error
Translate
Closure
Finding the Amount of Time Spent Jogging
A literal equation shows the relationship of two or more variables. In some cases, a literal equation needs to be solved for a specific variable. Coming back to the challenge presented at the beginning of the lesson, the time Maya spent jogging can also be found by solving the given formula for t. d=v * t It is known that Maya will run 150 meters at a speed of 2.5 meters per second.
a Determine how long it takes Maya to jog this distance by trying some values for some t-values until a true statement is obtained.
b Find a different way to solve the question.
Answer
a Table:
| Time (s) | Distance Traveled (m) |
|---|---|
| 10 | 25 |
| 20 | 50 |
| 30 | 75 |
| 40 | 100 |
| 50 | 125 |
| 60 | 150 |
It takes Maya 60 seconds to run 150 meters.
b See solution.
Hint
a Use multiples of 10 for t to make a table.
b Solve the formula for t and substitute the given values.
Solution
a It is known that Maya's speed is 2.5 meters per second. By the formula, the speed multiplied by the time equals the distance traveled. For different t-values, the distance traveled is shown in the table.
| Time (s) | Speed * Time | Distance Traveled (m) |
|---|---|---|
| 10 | 2.5 ( 10) = 25 | 25 |
| 20 | 2.5 ( 20) = 50 | 50 |
| 30 | 2.5 ( 30) = 75 | 75 |
| 40 | 2.5 ( 40) =100 | 100 |
| 50 | 2.5 ( 50) = 125 | 125 |
| 60 | 2.5 ( 60) = 150 | 150 |
Maya will run 150 meters in 60 seconds at a speed of 2.5 meters per second.
b The same result can be found by solving the formula for t and then substituting in the given values.
d=v* t
▼
Solve for t
DivEqn
.LHS /v.=.RHS /v.
d/v = v * t/v
CancelCommonFac
Cancel out common factors
d/v = v * t/v
SimpQuot
Simplify quotient
d/v = t
RearrangeEqn
Rearrange equation
t = d/v
Now, substitute d = 150 and v=2.5.
Share
Report error
Translate
Rewriting and Solving Literal Equations
Exercise 1.1
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["S","p"],"constants":[]},"rendered":false},"text":""},"formTextBefore":"B=","formTextAfter":null,"answer":{"text":["\\dfrac{S}{1+p}"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["S","B"],"constants":[]},"rendered":false},"text":""},"formTextBefore":"p=","formTextAfter":null,"answer":{"text":["\\dfrac{S}{B}-1"]}}
security
report
code
security
report
code
To isolate B, we will divide both sides of the formula by (1+p).
We will start by dividing both sides of the given formula by B. Then, we will use the inverse operations to isolate p.
security
report
code
E
Hint & Answer
S
Solution
{"type":"choice","form":{"alts":["A","B","C"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":2}
{"type":"choice","form":{"alts":["A","B","C"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":0}
security
report
code
security
report
code
Isolating R means that it should stand alone on one side of the equation. We can write the law in that form by dividing both sides by nT.
Therefore, R is equal to PVnT. This corresponds to C.
This time we will divide the formula by P to isolate V.
The volume V can be expressed as nRTP. This corresponds to A.
security
report
code
E
Hint & Answer
S
Solution
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["V","r"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":"h=","formTextAfter":null,"answer":{"text":["\\dfrac{V}{\\pi*r^2}"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":"About","formTextAfter":"cm","answer":{"text":["11"]}}
security
report
code
security
report
code
We need to solve the formula for h. Since h is multiplied by both π and r^2, we will divide the equation by π r^2.
The formula for h thus becomes h = Vπ r^2.
We will use the formula from the previous part and substitute the given values.
The Coca-Cola jar has a height about 11 centimeters.
security
report
code
E
Hint & Answer
S
Solution
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["a"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":"b=","formTextAfter":null,"answer":{"text":["17-a"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["g","t"],"constants":[]},"rendered":false},"text":""},"formTextBefore":"b=","formTextAfter":null,"answer":{"text":["t+g"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["p"],"constants":[]},"rendered":false},"text":""},"formTextBefore":"b=","formTextAfter":null,"answer":{"text":["\\dfrac{p}{7}"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x","y"],"constants":[]},"rendered":false},"text":""},"formTextBefore":"b=","formTextAfter":null,"answer":{"text":["xy"]}}
security
report
code
security
report
code
In the given equation, a is being added to b. Since the inverse operation of addition is subtraction, we can isolate the variable b by applying the Subtraction Property of Equality.
In this case, we have b and another variable g on the left-hand side of the equation. We can get rid of g by adding g to both sides.
Therefore, b is equal to the sum of t and g.
On the left side, b is multiplied by 7. To get rid of 7, we need to divide the equation by 7.
Now we have a y in the denominator of the fraction by. To isolate b, we multiply both sides of the equation by y.
security
report
code
E
Hint & Answer
S
Solution
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x","r"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":"P=","formTextAfter":null,"answer":{"text":["2x+2\\pi r"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["P","r"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":"x=","formTextAfter":null,"answer":{"text":["\\dfrac{1}{2}P- \\pi r","\\dfrac{P}{2}- \\pi r","\\dfrac{P-2\\pi r}{2}"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":"meters","answer":{"text":["140"]}}
security
report
code
security
report
code
The perimeter of the given figure is made up of two sides of a rectangle and two semicircles. The two sides each have a length of x, and the two semicircles make up a circle with a radius r. To find their length, let's substitute the radius into the formula for the circumference of a circle. C = 2 π r Now that we have the circumference of the circle, we can write the perimeter P of the entire figure, which is a sum of all the sides. P= x + x+ 2 π r Let's simplify the expression by adding like terms. P=x + x + 2 π r ⇕ P=2x+2 π r
In order to solve the formula for x, we need to isolate x on one of the sides.
The perimeter is given as 562 meters and the radius of the half circles is given as 45 meters. We can find x by substituting these values into the formula that we found in Part B.
When rounded to the nearest integer, the straight path x is about 140 meters long.
security
report
code
E
Hint & Answer
S
Solution
A truck travels on a highway at a rate of 108 kilometers per hour. How long does it take the truck to travel 162 kilometers?
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":"minutes","answer":{"text":["90"]}}
security
report
code
security
report
code
Let's begin with recalling the formula for the distance traveled. d=r t In this formula d represents the distance, r is the rate, and t stands for time. In our exercise we are given that the truck travels 108 kilometers per hour and we want to determine the time the truck needs to travel 25 miles. Let's first isolate t. d=r t ⇒ t = d/r We can now substitute 108 for r and 162 for d into the formula.
The truck needs 1.5 hours to travel 162 kilometers. To convert it into minutes, we will multiply the time by the conversion factor 60 min1 h.
The truck travels 162 kilometers in 90 minutes.
security
report
code
E
Hint & Answer
S
Solution
Rewriting and Solving Literal Equations
Recommended exercises
To get personalized recommended exercises, please login first.
Loading content
Edit Lesson
≥
>
■2
■▢
e▢
7
8
9
×
÷■1
=
=
√▢
▢√
∫▢▢
4
5
6
+
−
≤
<
log
ln
log▢
1
2
3
()
∣
sin
cos
tan
0
.
π
x
y
Loading content