1. Using Inductive Reasoning to Create Equations
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Chapter 1
1.
Using Inductive Reasoning to Create Equations
Inductive reasoning serves as a practical method for identifying patterns in numerical and visual data. This method is particularly useful in algebra and geometry. The lesson delves into the importance of variables, which are symbols used to represent unknown quantities. It also covers linear expressions and equations, which are foundational elements in algebra. These concepts are not just theoretical; they have real-world applications. For instance, understanding how to form equations can help in financial planning or in engineering problems. The lesson also emphasizes that equations often contain one or more unknown values called variables, which can be solved to make the equation true.
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| | 14 Theory slides |
| | 9 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Using Inductive Reasoning to Create Equations
Slide of 14
Mathematics can be used to model real-life situations. This is helpful to understand those situations and find solutions to problems that might arise in such scenarios. This lesson will discuss the concepts of variable, linear expression, equation, and linear equation.
Catch-Up and Review
Here is a recommended reading before getting started with this lesson.
Challenge
Organizing Tables in a Restaurant
A group of people at a restaurant is waiting for a table for a family dinner. Waiters began adding tables one by one until the entire group was seated as requested.
Is there an expression to represent the number of people in terms of the number of tables?
Discussion
Variable
Sometimes a quantity is unknown or its value may change. If this is the case, the best way to represent this quantity is using a variable.
A variable is a symbol used to represent an unknown quantity. Often, variables represent fixed but unknown numbers. Variables are usually denoted with letters such as x.
x+1=8
In this case, x is a variable in an equation. Solving this equation for x will identify the value of x that makes the equation true.
x+1=8⇔x=7
Alternatively, a variable can be used to represent a quantity that changes.
Izabella’s income varies depending on thenumber of hours she works. She receives afixed salary of $100 per week plus $4 per hour.
In this case, Izabella's weekly income could be different every week, depending on the number of hours she works. Therefore, the use of a variable is appropriate. Let x be the hours that Izabella works in a week. Her income can be written by adding the fixed $100 to the $4 per hour. Izabella’s Weekly Income4x+100
Discussion
Linear Expression
There are many real-life situations that can be modeled by combining numbers and variables in algebraic expressions. A special case of algebraic expressions are linear expressions.
Concept
Linear Expression
A linear term is an algebraic expression that includes a coefficient multiplied by a variable with an exponent of one. A linear expression is an expression that includes at least one linear term and any constant terms. No other type of terms may be included. The most common form of a linear expression is given below.
In this expression, a and b are real numbers, with a=0. To completely understand the definition of a linear expression, some important concepts will be be broken down. Consider the example linear expression x−5y+2.
| x−5y+2 | ||
|---|---|---|
| Concept | Explanation | Example |
| Term | Parts of an expression separated by a +or −sign. |
x, -5y, 2 |
| Coefficient | A constant that multiplies a variable. If a coefficient is 1, it does not need to be written due to the Identity Property of Multiplication. | 1, -5 |
| Linear Term | A term that contains exactly one variable whose exponent is 1. | x, -5y |
| Constant Term | A term that contains no variables. It consists only of a number with its corresponding sign. | 2 |
The following table shows some examples of linear and non-linear expressions.
| Linear Expressions | Non-linear Expressions |
|---|---|
| 3x | 5 |
| -5y+1 | 2xy−3 |
| 3x−21y+2 | x1−2 |
| πx+6y | 5x2+x−1 |
3x+4−2x+9−x=13
The expression 3x+4−2x+9−x looks like a linear expression, but the result after simplifying is 13, which is a constant and therefore not a linear expression.Example
Classifying the Parts of a Linear Expression
Ramsha, Tiffaniqua, Zosia, and some of their friends are spending the summer at math camp. One of their first lessons is about identifying parts of linear expressions. They are given the following table to fill in based on the linear expressions in the top row.
| 6x−1 | 2+x | -2x+3−9y | |
|---|---|---|---|
| Term(s) | |||
| x-term | |||
| y-term | |||
| Linear Term(s) | |||
| Coefficient(s) | |||
| Constant Term |
Help them fill in the table.
Answer
| 6x−1 | 2+x | -2x+3−9y | |
|---|---|---|---|
| Term(s) | 6x and -1 | 2 and x | -2x, 3, and -9y |
| x-term | 6x | x | -2x |
| y-term | - | - | -9y |
| Linear Term(s) | 6x | x | -2x and -9y |
| Coefficient(s) | 6 | 1 | -2 and -9 |
| Constant Term | -1 | 2 | 3 |
Hint
When there is no number in front of a variable, the coefficient is 1. The x- and y-terms are the terms that contain the variables x and y, respectively.
Solution
Start by finding the terms of each expression. Recall that the terms are the parts separated by addition or subtraction signs.
| 6x−1 | 2+x | -2x+3−9y | |
|---|---|---|---|
| Term(s) | 6x and -1 | 2 and x | -2x, 3, and -9y |
| x-term | |||
| y-term | |||
| Linear Term(s) | |||
| Coefficient(s) | |||
| Constant Term |
The x-term is the term that contains the variable x.
| 6x−1 | 2+x | -2x+3−9y | |
|---|---|---|---|
| Term(s) | 6x and -1 | 2 and x | -2x, 3, and -9y |
| x-term | 6x | x | -2x |
| y-term | |||
| Linear Term(s) | |||
| Coefficient(s) | |||
| Constant Term |
Similarly, the y-term is the term that contains the variable y. In this case, the first two expressions do not have a y-term.
| 6x−1 | 2+x | -2x+3−9y | |
|---|---|---|---|
| Term(s) | 6x and -1 | 2 and x | -2x, 3, and -9y |
| x-term | 6x | x | -2x |
| y-term | - | - | -9y |
| Linear Term(s) | |||
| Coefficient(s) | |||
| Constant Term |
The linear terms are the terms that contain only one variable, raised to the power of 1.
| 6x−1 | 2+x | -2x+3−9y | |
|---|---|---|---|
| Term(s) | 6x and -1 | 2 and x | -2x, 3, and -9y |
| x-term | 6x | x | -2x |
| y-term | - | - | -9y |
| Linear Term(s) | 6x | x | -2x and -9y |
| Coefficient(s) | |||
| Constant Term |
Next, the coefficients are the numbers that multiply a variable. When there is no number in front of a variable, the coefficient is 1.
| 6x−1 | 2+x | -2x+3−9y | |
|---|---|---|---|
| Term(s) | 6x and -1 | 2 and x | -2x, 3, and -9y |
| x-term | 6x | x | -2x |
| y-term | - | - | -9y |
| Linear Term(s) | 6x | x | -2x and -9y |
| Coefficient(s) | 6 | 1 | -2 and -9 |
| Constant Term |
Finally, the constant term is any term without a variable.
| 6x−1 | 2+x | -2x+3−9y | |
|---|---|---|---|
| Term(s) | 6x and -1 | 2 and x | -2x, 3, and -9y |
| x-term | 6x | x | -2x |
| y-term | - | - | -9y |
| Linear Term(s) | 6x | x | -2x and -9y |
| Coefficient(s) | 6 | 1 | -2 and -9 |
| Constant Term | -1 | 2 | 3 |
Pop Quiz
Identifying the Parts of a Linear Expression
Select the required parts of the given linear expressions.
Example
Modeling Distance Traveled Using Linear Expressions
Ramsha is on the second day of a hiking excursion at math camp. On the first day, she hiked from base camp to the first station.
Distance From the Base Camp1.5x+10
Here, x is the number of hours Ramsha walks on the second day. a What does the coefficient 1.5 represent in this context?
b What does the linear term 1.5x represent in this context?
c What does the constant term 10 represent in this context?
Answer
a The speed at which Ramsha walks.
b The distance Ramsha walks on the second day.
c The distance from base camp to the first station.
Hint
b The expression represents distance. Therefore, each term in the expression also represents distance.
c The constant term represents a fixed distance.
Solution
a Start by representing Ramsha's current position from the base camp in the diagram.
Note that 1.5x+10 represents a distance and the units of each term are miles. However, the units of x are hours. This gives a clue of the units of the coefficient 1.5.
In order to get rid of hours
and end up with miles,
the units of the coefficient must be miles per hour.
Therefore, the coefficient represents speed, or more precisely, the speed at which Ramsha walks.
b Since x represents the number of hours Ramsha has walked on the second day, then 1.5x represents the distance Ramsha has walked on the second day, starting at the first station.
c Finally, since 1.5x+10 represents the distance from base camp and 1.5x the distance walked on the second day, then the constant term 10 represents the distance Ramsha walked on the first day — that is, the distance from base camp to the first station.
Discussion
Inductive Reasoning
In the previous example, an expression modeling a real-life scenario was provided. However, most of the time such expressions are not provided and must instead be created. The use of inductive reasoning can help with this process.
Inductive reasoning is the process of finding patterns in specific observations and writing a conclusion or conjecture. Since the conjecture is based on observations, it might be false. For example, suppose an observer notices that all the birds around them are white. The observer might inductively reason that all birds in the world are white, which is not true.
Inductive reasoning is a practical method used in geometry and algebra for recognizing visual and numerical patterns. For instance, use the first three figures in the following diagram to guess the shape and the number of cubes in the fourth figure.
After observing the first three figures, it is reasonable to conclude that the number of cubes increases by 3 from one figure to the next — one on the top, one on the front, and one on the right. Since the figure starts with 1 cube and each step adds 3 cubes, an expression can be written to model the number of cubes in each figure.
1+3n
Here, n represents the number of steps after Figure 1. For example, Figure 2 is one step after Figure 1, so n=1. This expression allows the calculation of the number of cubes in any figure of the pattern. | n | Number of Cubes | |
|---|---|---|
| Figure 1 | 0 | 1+3⋅0=1 |
| Figure 2 | 1 | 1+3⋅1=4 |
| Figure 3 | 2 | 1+3⋅2=7 |
| Figure 4 | 3 | 1+3⋅3=10 |
| Figure 121 | 120 | 1+3⋅120=361 |
Example
Expressing the Cost of an Amusement Park Trip
At the end of the math camp, the students all visit an amusement park. Tiffaniqua did so well at camp that she won a coupon that allows her entire friend group to go on rides for x dollars per ride. By the end of the day, the group has gone on 11 rides and each person has eaten one hot dog.
Write an expression that represents the total amount of money the group paid.
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{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["k","d"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["\\dfrac{11x+kd}{k}","\\dfrac{11x+dk}{k}"]}}
Hint
The total amount of money paid is the sum of the amount paid for the 11 rides and the amount spent in hot dogs. The cost per person is the total cost divided by the number of people.
Solution
Start by writing what each letter represents.
xdk→cost of a ride→cost of a hot dog→number of people
Thanks to the coupon, the entire group pays x dollars per ride. Since they went on 11 rides, they paid 11x dollars.
Cost of the rides: 11x
The cost of the hot dogs must be added to the previous amount. The price of a hot dog is d dollars. Since there are k people, the total cost is kd dollars.
Cost of the hot dogs: kd
Finally, the expression representing the total amount the group paid is the sum of the amount paid in rides and the amount paid in hot dogs.
Total cost: 11x+kd
If the group wants to divide the cost evenly, then the total cost must be divided by the total number of people. In this case, the expression above must be divide by k.
Cost per person: k11x+kd
Discussion
Equations
In the previous two examples, each situation was represented by an algebraic expression. There are some cases where writing an expression is not enough to model a situation. In such cases, an equation might be required.
Concept
Equation
An equation is a type of mathematical relation that indicates that two quantities are equal. Equations often contain one or more unknowns values called variables. Some examples are shown below.
Solving an equation with a variable means to find the value or values of each variable that make the equation true.
Discussion
Linear Equation
Equations always contain an equals sign, while expressions do not. In fact, an equation can be seen like a statement that connects two expressions with an equals sign.
When learning about equations, a good first step is to start with linear equations, which involve only one variable.
Concept
Linear Equation
A linear equation is an equation with at least one linear term and any number of constants. No other types of terms may be included. Linear equations in one variable have the following form, where a and b are real numbers and a=0.
ax+b=0orax=b
Example
Money Saving
Zosia wants to go shopping after camp, but she realizes that she spent too much money at the amusement park. She decides to start saving money and go shopping when she has $25 saved up. She puts $15 in her piggy bank and plans to add $0.50 every day.
Let t be the number of days Zosia has been saving money. Write an equation that models the moment when her savings reaches $25.
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Hint
Zosia's savings can be modeled by a linear expression of the form at+b, where a and b are real numbers.
Solution
First, write an expression modeling Zosia's savings. To do this, note that the amount of money in the piggy bank increases by a constant amount every day. Therefore, it can be modeled by a linear expression.
at+b
The coefficient a is the rate of change and the constant b is the initial amount. Since Zosia starts with $15 and saves $0.50 every day, it can be said that a=0.5 and b=15.
0.5t+15
The expression above represents the money in the piggy bank after t days. Finally, the expression above has to be equated to something else.
0.5t+15=?
To figure out what comes on the right-hand side, translate the equation above into words.
The money in the piggy bank after t days is equal to …
Since the aim is to model the moment in which Zosia reaches $25, the right-hand side of the equation is 25. 0.5t+15=25
Example
Counting the Candy From the Store
Now that she has money saved up, Zosia is finally going to the mall. At the candy store, she buys 3 bags containing the same number of candies. The store also gives her 2 free candy samples. When she arrives home, she opens the bags and counts a total of 17 candies.
Let x be the number of candies in each bag. Write an equation that models the total amount of candies Zosia got from the store.
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Hint
What is the coefficient for the x-term in this situation?
Solution
First, write an expression that models the number of candies from the store. To do so, note that the total number of candies depends on the number of candies in each bag, the number of bags, and the number of extra candies. A linear expression can be used.
ax+b
The variable x represents the number of candies in each bag. The coefficient a is the number of bags, and the constant b is the number of free candies. Then, a=3 and b=2.
3x+2
The expression above represents the total number of candies after opening the 3 bags with x candies each. Since an equation is required, the expression above has to be equated to something else.
3x+2=?
Zosia counted 17 total candies. Therefore, the right-hand side of the equation is 17.
3x+2=17
Closure
Writing Algebraic Expressions to Model Real-Life Situations
This lesson discussed linear expressions and linear equations, as well as how to use inductive reasoning to create them. Countless real-life situations can be modeled by algebraic expressions and equations. Consider again the challenge presented at the beginning of the lesson. 
As the waiters add more tables for the group of people at the restaurant for the family dinner, they begin to notice a pattern. Help the waiters by writing an algebraic expression that represents the number of people they can seat in terms of the number of tables x. 
A linear expression can be used to model this situation.
Therefore, it can be said that b=2. With this, the expression is complete.
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Hint
Notice how the number of people sat increases with each table added.
Solution
Begin by making sense of the situation. At least 1 table is needed for the family dinner. In this case, 4 people can be seated.
For each subsequent table, two more people are seated.
ax+b
Here, a=2 because adding a table will add two more seated people. It can be noted that there will be always 2 people at the ends of the big table.
Number of people=2x+2
Using Inductive Reasoning to Create Equations
Exercises
Translate each sentence into an equation.
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{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["a"],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["3-5a=7"]}}
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Every equation has an equals sign and values or expressions on either side of it. Key phrases such as is,
is equal to,
and equals
tell us about the placement of the equals sign.
The sum of5and two timesy = 2
On the left-hand side, we have two key phrases, sum
and times.
These phrases tell us the operations that will be used in our equation. Times
indicates multiplication, while sum
indicates addition.
The sum of 5 and two times y
5 + 2 y
On the right-hand side, we have only 2, so we can now complete the equation.
5 + 2 y = 2
Let's begin by placing the equals sign according to the sentence.
Five timesaless than3 = 7
On the left-hand side, we have two key phrases, times
and less than.
These phrases tell us the operations that will be used in our equation. Times
indicates multiplication, while less than
indicates subtraction.
Five times a less than 3
3 - 5 a
On the right-hand side, we have only 7, so we can now complete the equation.
3 - 5 a = 7
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Translate each sentence into an equation.
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{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["a","b"],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["3(a+b)=9"]}}
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Every equation has an equals sign and values or expressions on either side of it. Key phrases such as is,
is equal to,
and equals
tell us about the placement of the equals sign.
The sum ofpand4 = two timesp
On the left-hand side, we have the key phrase sum.
This phrase tells us which operation that will be used in our equation — in this case, addition.
The sum of p and 4
p + 4
On the right-hand side, we have the key phrase times.
This indicates multiplication.
two times p
2 p
We can now complete the equation.
p + 4 = 2 p
Let's begin by placing the equals sign according to the sentence.
Three times the sum ofaandb = 9
On the left-hand side, we have two key phrases, times
and sum.
The word times
indicates multiplication, while sum
indicates addition. In this case, the multiplication is of the whole sum of a and b, so parentheses must be added.
Three times the sum of a and b
3 ( a + b )
We have only 9 on the right-hand side, so we can now complete the equation.
3 ( a + b ) = 9
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Consider the following equation.
3x+5=9
Which of the following sentences is a translation of the equation?
ABCDFive times three x equals 9.Three times the sum of x and 5 is equal to 9.Three times x increased by 5 equals 9.Five increased by 3 equals 9 times x.
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Every equation has an equals sign and values or expressions on either side of it. The given equation is no different, so let's begin our translation by using phrases such as is,
is equal to,
and equals
to express equality.
3x+5 = 9
3x+5 equals9
Now, to translate the left- and right-hand sides of this equation, let's break down the expressions and verbally express the individual terms and factors.
| Algebraic Term | Verbal Translation |
|---|---|
| 3 x | Three times x |
| + | increased by |
| 5 | 5 |
| = | equals |
| 9 | 9 |
| Three times x increased by 5 equals 9 | |
The answer is option C. Note that because there are often several ways to express certain mathematical operations, this answer is only one of many correct ways to verbally express this equation.
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Consider the following equation.
7(a+2)=b−3
Which of the following sentences is a translation of the equation?
ABCDSeven times the sum of a and 2 is equalto b decreased by 3.The sum of seven times a and 2 is equalto b minus 3.a plus 2 times 7 equals b decreased by 3.Seven and a times 2 is equal to b negative 3.
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Every equation has an equals sign and values or expressions on either side of it. The given equation is no different, so let's begin our translation by using phrases such as is,
is equal to,
and equals
to express equality.
7(a+2) = b-3
7(a+2) is equal to b-3
Now, to translate the left- and right-hand sides of this equation, let's break down the expressions and verbally express the individual terms and factors.
| Algebraic Term | Verbal Translation |
|---|---|
| 7( a + 2) | Seven times the sum of a and 2 |
| = | is equal to |
| b | b |
| - | decreased by |
| 3 | 3 |
| Seven times the sum of a and 2 is equal to b decreased by 3 | |
The answer is option A. Note that because there are often several ways to express certain mathematical operations, this answer is only one of many correct ways to verbally express this equation.
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Translate the sentence into a formula.
The perimeter P of a regular hexagon is 6 times the length s of each side.
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Our formula will be expressed as an equation. Every equation has an equals sign and values or expressions on either side of it. Key phrases such as is,
is equal to,
and equals
tell us about the placement of the equals sign.
The perimeter P of a regular hexagon = 6times the lengths of each side.
On the left-hand side, we only have the perimeter P.
The perimeter P of a regular hexagon
P
Now we will take care of the right-hand side.
6times the length s of each side
6times s
We also have a single key word on the right-hand side, times.
This indicates multiplication.
6times s
6 s
Putting these sides together, we have a complete formula.
P = 6 s
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Using Inductive Reasoning to Create Equations
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