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Relationships Between Quantities using Equations and Inequalities

Solving Multi-Step Equations in One Variable

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When it comes to modeling daily life situations using equations, it is common to end with equations that require more than one step to be solved. For such cases, it is convenient to know which properties can be applied to solve the obtained equations. To learn about it, different types of equations will be set and solved along this lesson.

Catch-Up and Review

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

Challenge

Writing Equation for Number of Tickets Bought

Mark wants to buy tickets online to play paintball with his friends next weekend. On the website, he finds the following price list.
Course 1, ticket price $15; Course 2, ticket price $18; Course 3, ticket price $25; Course 4, ticket price $30. Apart from the Course price, the processing fee per ticket is $1.5, and the service charge is $4
Note that the service charge is per transaction, not per ticket.
a Mark and his friends decide to choose Course and pay Let be the number of tickets bought. Write an equation in terms of that models this situation.
b Mark and his friends plan to go again next month but on Course They were told that when the group has more than eight people, the processing fee of five tickets is free. Write an equation modeling the situation in which the group gets the discount and pays
c How many tickets did Mark buy for his first visit? How many tickets will Mark buy next month?

Challenge

Matching Equivalent Expressions

Two numerical expressions are equivalent if one of the expressions is obtained from the other by applying properties of addition and multiplication. Knowing this, match each expression in the left column with the equivalent expression.

Discussion

Properties of Addition and Multiplication

Apart from the Properties of Equality, the properties of addition and multiplication are also heavily used when rewriting and simplifying expressions or equations. Thus, it is convenient to know these properties. The commutative property of each operation will be discussed first.

Rule

Commutative Property of Addition

The order in which two or more terms are added does not affect the value of the sum. That is, the addends can be written in any order.

For example, adding to produces the same result as adding to In both cases, the sum is This property also applies to the sum of more than two terms.
Since the Commutative Property of Addition is an axiom, it does not need a proof.

Rule

Commutative Property of Multiplication

The order in which two or more factors are multiplied does not affect the value of the product. That is, the multiplicands can be written in any order.

For example, multiplying by produces the same result as multiplying by In both cases, the product is This property also applies to the product of more than two terms.
Since the Commutative Property of Multiplication is an axiom, it does not need a proof.

As seen, the commutative property says that the order in which two or more terms are added or multiplied will not change the resulting sum or product. Next, the associative property of addition and multiplication will be presented.

Rule

Associative Property of Addition

The way three or more terms are grouped when added does not affect the value of the sum.

For example, consider the sum Grouping and adding it to produces the same result as grouping and adding it to
Since the Associative Property of Addition is an axiom, it does not need a proof.

Rule

Associative Property of Multiplication

The way three or more factors are grouped when multiplied does not affect the value of the product.

For example, consider the product Grouping and multiplying it by produces the same result as grouping and multiplying it by
Since the Associative Property of Multiplication is an axiom, it does not need a proof.
These four properties can be used, together or separately, to rewrite an expression in a different but equivalent way.

Pop Quiz

Recognizing Equivalent Expressions

Two expressions are equivalent when one of them is obtained by applying properties of addition and multiplication to the other. For each pair of given expressions, determine whether they are equivalent. In the affirmative case, name the property that transforms one expression into the other.

A pair of random expressions is shown and it must be determined if they are equivalent

Discussion

Substituting Expressions

When solving equations, there are instances in which some parts of an expression can be substituted with an equivalent expression. As an example, consider the following equation.
Since equals the second term on the left-hand side can be substituted with
The previous simplification is able thanks to the Substitution Property of Equality, which also guarantees that the resulting equation and the initial are equivalent.

Rule

Substitution Property of Equality

If two real numbers are equal, then one can be substituted for another in any expression.

If then can be substituted for in any expression.

Since the Substitution Property of Equality is an axiom, it does not need a proof.

Example

Temperature Conversion

This morning in science class, Tadeo and Magdalena learned how to convert degrees Celsius to Fahrenheit using the following formula.
Later, while eating lunch in the schoolyard, they saw that the thermometer indicated a temperature of They wondered what the temperature would be in Celsius. To make the conversion, they used the formula they learned in the morning. However, they got different results.
Tadeo got 23 C degrees and Magdalena got 25 C degrees.
Who is correct?

Hint

Apply the Substitution Property of Equality to determine who got the correct answer. Start by substituting for into the conversion formula. Then, substitute the values Tadeo and Magdalena found for If a true statement is obtained, the corresponding value is a solution; otherwise, it is not.

Solution

Tadeo and Magdalena want to convert into degrees Celsius. Therefore, by the Substitution Property of Equality, can be substituted for in the conversion formula.
Again, by the Substitution Property of Equality, the values found by Tadeo and Magdalena will be substituted for in the above equation one at a time.
After substituting each value and simplifying, if a true statement is obtained, the substituted value is a solution of the equation. Otherwise, it is not. First, will be substituted.
Substitute for and simplify
A false statement was obtained, which suggests that Tadeo's answer is not correct. Next, substitute Magdalena's answer.
Substitute for and simplify
A true statement was obtained, which implies that Magdalena is correct in saying that is the same as

Discussion

Distributive Property of Multiplication

In addition to the commutative and associative properties, multiplication has another useful property that helps in the process of simplifying expressions with parentheses. This property is called the Distributive Property.

Rule

Distributive Property

Multiplying a number by the sum of two or more addends produces the same result as multiplying the number by each addend individually and then adding all the products together.

Note that the factor outside the parentheses is multiplied, or distributed, to every term inside. The Distributive Property is used to simplify expressions with parentheses.
a(b+c)=ab + ac
Since the Distributive Property is an axiom, it does not need a proof.
Note that, by the Symmetric Property of Equality, the Distributive Property can also be used to take out a common factor in two or more addends.
In some cases, the right-hand side expression is more useful than the one on the left.

Discussion

Solving Multi-Step Equations

When solving equations, sometimes more than one step is needed. Both the number of steps and the operations required depend on the complexity of the given equation. For example, consider the following pair of equations.

The general idea is to simplify both sides of an equation and then isolate the variable on one side of the equation.

Solving Equation (I)

Equation (I) contains a fraction and parentheses. In this case, it would be better to clear the fraction first.
1
Multiply to Clear Any Fractions
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If there are fractions in the equation, start by multiplying the entire equation by the least common denominator. In this case, multiply both sides of Equation (I) by
By applying the Distributive Property, distribute one the left-hand side.
Then, cross out the common factors and perform the required multiplications. After this, Equation (I) is simplified into an equivalent equation with no fractions.
2
Clear Parentheses and Combine Like Terms on Each Side
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Apply the Distributive Property again to clear the parentheses on the left-hand side of the equation. In this case, distribute to each term inside the parentheses. Then, combine like terms.
3
Isolate the Variable
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Apply the Properties of Equality to isolate the variable on one side of the equation. In this case, first apply the Subtraction Property of Equality and then apply the Division Property of Equality.
Consequently, is the solution of Equation (I).

Solving Equation (II)

In this equation, the variable is on both sides of the equation. Therefore, solving the equation will require the additional step of collecting the variables on one side of the equation.
1
Multiply to Clear Any Fractions
expand_more

Since Equation (II) has no fractions, this step can be omitted.

2
Clear Parentheses and Combine Like Terms on Each Side
expand_more
Apply the Distributive Property to clear the parentheses on each side of the equation. In this case, there are no parentheses to clear, so instead, start by combining like terms.
3
Collect the Variable on One Side of the Equation
expand_more
Apply the Properties of Equality to collect all the variables on one side of the equation. In this case, add to both sides of the equation to group the terms on the left-hand side.
4
Isolate the Variable
expand_more
Apply the Properties of Equality again to isolate the variable on one side of the equation. In this case, add to both sides and then divide both sides of the equation by
In conclusion, is the solution of Equation (II).
Note that the operations used to solve Equations (I) and (II) may not be the same as those required to solve a different equation.

Example

Finding the Usual Price Based on the Discounted Price

At Ali's fruit store, strawberries are on sale! Today they are cheaper per pound than usual. Magdalena stopped at the store on her way home from school and bought eight pounds of strawberries. She paid a total of

Ali is inviting people to come and buy strawberries.
a Let be the usual price of one pound of strawberries. Write an equation, in terms of modeling Magdalena's purchase.
b If Tadeo bought six pounds of strawberries two weeks ago, how much did he pay?

Hint

a Today, the special price per pound of strawberries is dollars. Multiply this price by the number of pounds bought by Magdalena and equate it to the total amount she paid.
b Because Tadeo bought strawberries two weeks ago, he paid the usual price for them. Solve the equation set in Part A to find the usual price. Then, multiply it by the number of pounds bought by Tadeo.

Solution

a Let be the usual price of one pound of strawberries. According to Ali's offer, the special price is cheaper per pound than the usual price. Therefore, the special price equals the usual price minus
With this special price, Magdalena paid for eight pounds of strawberries. Consequently, multiplied by the is equal to the Using this information, an equation in terms of can be established.
This equation can be used to determine the usual price of a pound of strawberries.
b Since Tadeo bought the strawberries two weeks ago, he paid the usual price. Therefore, start by finding the usual price by solving the equation set in Part A.
First, apply the Distributive Property to clear the parentheses. Then, apply the Properties of Equality to isolate the variable.
This means that the usual price of one pound of strawberries is To find how much Tadeo paid, multiply — the number of pounds he bought — by the usual price.
Consequently, last week Tadeo paid for six pounds of strawberries. Note that Tadeo paid the same as Magdalena, but he bought pounds fewer.

Example

Solving an Equation With Variables on Both Sides

After dinner, Magdalena works on a model of a castle using cardboard for a history project. She wants the walls of the ramparts to be inches wide. Additionally, she wants the value of the perimeter of the wall, in inches, to be the same as the value of its area, in square inches.

The rampart of a castle. The rampart is a rectangle with dimensions 7 by x, and at the top, two small rectangles are cut. The two small rectangles are 2 by 1.
How high should the wall be?

Hint

The area of the wall is minus the area of the two small rectangles at the top. Equate the area expression with the perimeter of the wall. Then, solve the equation for

Solution

Since Magdalena wants the wall to have the same perimeter and area, regardless of the units of measure, start by finding the perimeter of the wall. To do so, add all the side lengths of the wall. Notice that the shorter sides of the small rectangles have a length of inch. Knowing this, the length of the top of the wall can be found.
The right-hand side of the wall has the same length as the left-hand side of the wall — that is, both have a length of inches. Adding these two lengths to the base length, inches, and to the length of the top of the wall, the perimeter of the wall will be obtained.
Substitute for and simplify
To find the wall's area, note that it is made by cutting two small rectangles from the bigger rectangle, whose dimensions are by
Cutting the two small rectangles from the big one
Consequently, the wall's area is equal to the area of the big rectangle minus the area of the two small rectangles. The big rectangle area is and the area of each small rectangle is square inches.
Since the perimeter and area of the wall must be the same, equate and
To find the wall's height, solve the equation for Since the equation has variables on both sides, start by collecting them on one side of the equation.
Therefore, the wall should be inches high.

Pop Quiz

Solving Different Equations

Solve the given equation for the indicated variable. If necessary, round the answer to two decimal places.
Random Equations. Some include variables on both sides.
After learning about the properties covered in this lesson, consider again the challenge presented at the beginning.

Closure

Finding the Number of Tickets Bought

At the beginning of the lesson, Mark was buying some paintball tickets. He found the following prices online, where the service charge is per transaction, not per ticket.
Course 1, ticket price $15; Course 2, ticket price $18; Course 3, ticket price $25; Course 4, ticket price $30. Apart from the Course price, the processing fee per ticket is $1.5, and the service charge is $4
a Write an equation modeling the situation in which Mark and his friends chose Course got no discount, and paid
b Write an equation modeling the situation in which Mark and his friends chose Course got the discount, and paid
c How many tickets did Mark buy in the first situation? How many tickets did Mark buy in the second?

Answer

a Example Equation:
b Example Equation:
c Mark bought tickets in the first situation and in the second.

Hint

a Find the amount paid only for the tickets, without fees. Next, find the amount paid in fees. Then, add these two costs to the service charge. Finally, equate the resulting expression with the actual money paid.
b Find the amount paid only for the tickets, without fees. Next, find the amount paid in fees. Remember, five of the tickets paid no fee. Then, add these two costs to the service charge. Finally, equate the resulting expression with the actual money paid.
c Solve the corresponding equation for Use the Distributive Property, the Associative Property of Addition, the Substitution Property of Equality, and the Properties of Equality.

Solution

a Start by listing the prices corresponding to Course
Course
Ticket price
Processing fee per ticket
Service charge
Let be the number of purchased tickets. Then, the amount paid only for the tickets without any fees will be the product of and
Since the group did not get a discount, they paid the processing fee for all the tickets bought. To find the amount paid only in fees, multiply the price of the processing fee by
They also had to pay a service charge, which is no matter the number of tickets bought. The sum of the three costs together will give an expression representing the total cost of purchasing tickets for Course
Finally, it is said that they paid a total of Therefore, equate the previous expression to

This equation models the situation in which Mark and his friends chose Course got no discount, and paid

b This time, the group decided to play on Course Recall the prices corresponding to Course
Course
Ticket price
Processing fee per ticket
Service charge
Let be the number of purchased tickets. Since the price per ticket is the amount paid only for the tickets — without any fees — will be the product of the price per ticket and
Since the group received a discount this time, they did not pay the fee for five of the tickets bought. Thus, to find the amount paid in fees, multiply the processing fee by
They also paid a service charge, which is no matter the number of tickets bought. By adding the three costs, an expression representing the money paid for the purchase of tickets for Course will be found.
Finally, it is said that they paid a total of Therefore, equate the previous expression to

This equation models the situation in which Mark and his friends chose Course got a discount, and paid

c To find the number of tickets bought by Mark the first time, the equation set in Part A will be solved for
Solve for
Consequently, the first time Mark bought tickets. This confirms why he did not get the discount. To find the number of tickets he bought the second time, the equation set in Part B will be solved for
Solve for
In conclusion, Mark bought tickets the second time.
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