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Real-life situations modeled by an equation include a coefficient of the variable which can be a rational number. Both sides of the equation are multiplied or divided by a number to isolate the variable. This lesson begins its focus on solving equations involving multiplication and division. Then, the focus shifts to how to write these equations when modeling real-life problems.

### Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

## A Vinyl Record Collection

Diego is going through some old stuff in storage. He discovers an entire rack of vinyls! There are in total and they belong to his dad. Diego asks his father how long this collection took to gather. a Write an equation in terms of that represents the situation.
b Solve the equation to find the time it took Diego's father to gather his collection of records.

## Multiplication Property of Equality

Multiplication and division are inverse operations. They can be used to solve equations by the following properties of equality.

## Multiplication Property of Equality

Given an equation, multiplying each side of the equation by the same number yields an equivalent equation. Let and be real numbers.

If then

The Multiplication Property of Equality is an axiom, so it does not need a proof. This property is one of the Properties of Equality that can be used when solving equations. Consider the following example.
Here, by multiplying both sides of the equation by the variable was isolated and the solution of the equation was found.

## Division Property of Equality

Dividing each side of an equation by the same nonzero number yields an equivalent equation. Let and be real numbers.

If and then

The Division Property of Equality is an axiom, so it does not need a proof to be accepted as true. This property is one of the Properties of Equality that can be used when solving equations.
As can be observed, by dividing both sides of the equation by the variable was isolated and the solution of the equation was found.

## Vinyl Collection Stored in Boxes

a Diego's father stored his vinyl collection in five boxes. Each box contains the same number of records and, in total, there are vinyl records in the collection. This situation can be represented by an equation.
Here, represents the number of records in one box. Solve this equation to find the number of records in each box.
b Diego's grandfather also found some records. Diego wants to put an equal number of those records grandpa found into each of the five boxes. He notices that he would have to put seven records into each box to do that. The following equation represents the situation.
In this equation, represents the number of records found by Diego's grandfather. Solve this equation to find the number of records Diego's grandfather found.

### Solution

a When solving equations in one variable, the variable is isolated on one side of the equation. This can be done by using inverse operations because inverse operations undo each other. Consider the given equation.
Here, the variable is multiplied by The inverse operation of multiplication is division, so each side of the equation is divided by to isolate The reason why this operation can be done is the Division Property of Equality, which ensures that both sides of the equation remain equal.
The solution to the given equation is The variable represents the number of records in each box, so each box contains records. The answer can be checked by substituting for in the equation.
Substituting for into the equation results in a true statement. This confirms that is the correct solution.
b Recall that the variable is isolated on one side when solving equations in one variable. Inverse operations play a role in isolating the variable because they undo each other. Consider the given equation.
In this equation, the variable is divided by The inverse operation of division is multiplication, so the Multiplication Property of Equality is used to multiply each side of the equation by The solution to the given equation is Here, represents the number of records found by Diego's grandfather. This means that Diego's grandfather found records. The solution can be substituted for in the equation to check the answer.
Substituting for into the equation results in a true statement. This means that is the correct solution. Diego's grandpa dances in celebration of Diego's math skills. ## Solving Equations With Rational Coefficients

The Multiplication Property of Equality can be used instead of the Division Property of Equality anytime when solving an equation with a coefficient that is a rational number. Consider an equation in the form
The coefficient is a fraction. For that reason, the equation can be solved by multiplying both sides by the reciprocal of As an example, consider the following equation.
The coefficient next to the variable is Interchange the numerator and denominator to find its reciprocal. The reciprocal of is so the equation can be solved by multiplying both sides by
Simplify

## Vinyl Records, Rotations, and Song Length

Diego finds himself wondering about the speed at which a record rotates. His father tells Diego that the record rotates times while playing a song on the album. Diego later reads online that a record rotates of a full rotation every second it plays. Diego combines these two pieces of information to write the following equation.
Here, represents the length of the song in seconds. Solve the equation to find the length of the song.

### Hint

Identify the coefficient of the variable. Multiply both sides of the equation by the reciprocal of the coefficient.

### Solution

Consider the given equation.
Here, the variable is and its coefficient is the fraction Since it is a fraction, isolate the variable by multiplying both sides of the equation by the reciprocal of that fraction. Recall that the reciprocal of a fraction is found by interchanging the numerator and denominator. The reciprocal of is Now, multiply the equation by
Simplify
The solution to the equation is which means that the song lasts seconds.

## Solving Equations

Solve the equations using the Multiplication Property of Equality or the Division Property of Equality. If necessary, round answers to two decimal places. ## Writing Equations

Many real-life situations can be algebraically modeled by equations. These equations can involve a variable that represents an unknown quantity. Consider modeling the following situation.

 Diego categorizes his father's record collection by genre. He finds out that there are different genres, each with the same number of records. The collection contains records. How many records are there in each genre?
This situation can be described in one sentence as follows.
Here, the unknown quantity is the number of records in each genre. Let be the variable representing this unknown quantity. Then, the verbal sentence can be translated into an algebraic equation.
The equation can now be solved to find the number of records in each genre. Use the Division Property of Equality.
Since represents the number of records in each genre, it can be concluded that in Diego's father's collection, there are records in each genre.

## Earning to Fix the Turntable

Diego's father's old turntable is broken. Diego is so eager to listen to the records that he decides to make some money to buy the replacement parts. The parts that Diego needs to buy costs Diego's neighbor offers him per walk to walk her dog. a Write an equation for the number of times Diego has to walk his neighbor's dog to earn enough money to buy the parts he needs.
b Solve the equation.
c Which of the following is a true statement?

### Hint

a Assign the variable to the unknown quantity.
b Use the Properties of Equality to solve the equation written in Part A.
c What does the variable represent?

### Solution

a The number of times Diego has to walk the dog is represented by the variable Diego gets per walk, so the total amount he earns is equal to He wants to save so the amount earned should be equal to By combining all this information, an equation can be written as follows.
This equation models the given situation.
b Solving an equation in one variable means isolating the variable on one side.
In this case, the variable is multiplied by Since the inverse operation of multiplication is division, the Division Property of Equality is used to isolate Now divide both sides of the equation by
The solution to the equation is
c The variable in the equation from Part A represents the number of times Diego needs to walk his neighbor's dog to earn enough money for the parts he needs to repair the broken turntable. The calculations show that This means that Diego needs to walk the dog exactly times to earn enough.

## Job Well Done

The dog Diego walked loved him so much — just look at the two of them! His neighbor, the dog owner, was so impressed that affter walks, she decided to pay Diego handsomely. Diego received for each walk.

a Write an equation for the amount of money Diego got in total.
b Solve the equation.
c Which of the following is a true statement?

### Hint

a Assign the variable to the unknown quantity.
b Use the Properties of Equality to solve the equation written in Part A.
c What does the variable represent?

### Solution

a The total amount of money Diego got for walking the dog is represented by the variable Diego walked the dog times, so the amount he earned for one walk is The neighbor paid Diego per walk, so the amount earned for one walk should be equal to Now, an equation can be written.
Note that different equations, equivalent to this one, can also be used to model this situation. An example of such an equation is
b Isolate the variable on one side to solve the equation.
Notice that is divided by This means that the Multiplication Property of Equality can be used to isolate because division and multiplication are inverse operations. The solution to the equation is
c The variable in the equation from Part A represents the total amount of money Diego earned walking his neighbor's dog. The calculations show that This means that Diego earned exactly
It is safe to say that Diego earned enough to fix the turntable. Play that jam! ## How Long Did It Take To Compile the Collection?

The challenge presented at the beginning of the lesson can be solved by writing an equation that models the situation and then solving the equation. It stated that Diego's father collected vinyl records and bought records every month. a Write an equation in terms of that represents the situation.
b Solve the equation to find the time it took Diego's father to gather his collection of records.

### Hint

a Assign a variable to the number of months.

### Solution

a When modeling a real-life situation with an equation, the first step is to assign a variable to the unknown quantity. In this case, the unknown quantity is the number of months it took Diego's father to gather his collection, so let it be
Diego's father bought records every month. Then, the number of records Diego's father collected in months is There are records in the collection, so must be equal to
Diego figured out the equation!
b The variable must be isolated to solve the equation. In this case, is multiplied by Divide both sides of the equation by to isolate Recall that dividing both sides by the same number does not change the solution to the equation because of the Division Property of Equality.
The solution to the equation is The variable represents the number of months it took Diego's father to gather his collection. This means it took months.

### Extra

Equations With Decimal Coefficients
On a final note, consider an equation where the coefficient is a decimal.
Equations where the coefficient is a decimal can be solved the same way as equations where the coefficient is an integer or a fraction. In this case, this means dividing both sides of the equation by the coefficient
Similarly, when solving equations where the variable is divided by a decimal, or a decimal is added or subtracted from the variable, we use the same methods we would when solving an equation with integers or fractions.
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