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3. Solving Multi-Step Equations in One Variable
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Chapter 1
3. 

Solving Multi-Step Equations in One Variable

In algebra, multi step equations hold significant importance. These equations often necessitate multiple operations to derive the solution. The lesson elaborates on the nuances of tackling such equations, emphasizing the pivotal role of properties of addition and multiplication. For instance, the commutative property indicates that the sequence in which numbers are added or multiplied doesn't alter the outcome. The distributive property, on the other hand, assists in breaking down expressions with parentheses. Everyday scenarios, such as determining the total cost of items or calculating discounts, frequently involve these multi step equations. By grasping the techniques showcased, individuals can proficiently address algebraic challenges and implement them in real-world situations.
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13 Theory slides
13 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
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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. In other words, 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
Substitute

MoveRightFacToNum

Multiply

Multiply

CalcQuot

Calculate quotient

AddTerms

Add terms

A false statement was obtained, which suggests that Tadeo's answer is not correct. Next, substitute Magdalena's answer.
▼
Substitute for and simplify
Substitute

MoveRightFacToNum

Multiply

Multiply

CalcQuot

Calculate quotient

AddTerms

Add terms

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 the equation and then isolate the variable on one side of the equation. This is usually done by collecting all the variable terms on one side of the equations and combining them. Then, the operations applied to the variable are undone in reverse order.

Solving Equation (I)

Equation (I) contains a fraction and parentheses.
1
Clear Parentheses and Combine Like Terms on Each Side
expand_more
Apply the Distributive Property 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 on the left-hand side.
Distr

Distribute

FracMultDenomToNumber

AddTerms

Add terms

2
Isolate the Variable
expand_more
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.
SubEqn

SubTerms

Subtract terms

The coefficient of the variable is a fraction. The Multiplication Property of Equality can be used to multiply both sides of the equation by the reciprocal of the coefficient.
MultEqn

CommutativePropMult

Commutative Property of Multiplication

MultFracByInverse

OneMult

MoveLeftFacToNum

Multiply

Multiply

CalcQuot

Calculate quotient

As such, is the solution of Equation (I).

Solving Equation (II)

In this equation, the variable is on both sides of the equation.
Solving the equation will require the additional step of collecting the variables on one side of the equation.
1
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. There are no parentheses to clear in this equation, so start by combining like terms instead.
AddTerms

Add terms

2
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.
AddEqn

CommutativePropAdd

Commutative Property of Addition

AddTerms

Add terms

3
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, then divide both sides of the equation by
AddEqn

AddTerms

Add terms

DivEqn

CalcQuot

Calculate quotient

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.
Distr

Distribute

Multiply

Multiply

AddEqn

DivEqn

CalcQuot

Calculate quotient

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.
Substitute

Multiply

Multiply

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.
AddTerms

Add terms

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
Substitute

CommutativePropAdd

Commutative Property of Addition

AssociativePropAdd

Associative Property of Addition

AddTerms

Add terms

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.
SubstituteII

,

Multiply

Multiply

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.
SubEqn

CommutativePropAdd

Commutative Property of Addition

SubTerms

Subtract terms

AddEqn

AddTerms

Add terms

DivEqn

CalcQuot

Calculate quotient

RearrangeEqn

Rearrange 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
AssociativePropAdd

Associative Property of Addition

AddTerms

Add terms

SubEqn

DivEqn

CalcQuot

Calculate quotient

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
Distr

Distribute

Multiply

Multiply

CommutativePropAdd

Commutative Property of Addition

AssociativePropAdd

Associative Property of Addition

AddSubTerms

Add and subtract terms

AddNeg

AddEqn

DivEqn

CalcQuot

Calculate quotient

In conclusion, Mark bought tickets the second time.



Level 1
Level 2
Level 3

Solve each equation for

We start by multiplying both sides of the equation by 7 to get rid of the denominator.

Now, we perform the required operations to simplify both sides of the equation.

Finally, we isolate the variable by applying inverse operations.

The solution to the equation is x=3.


In this equation, we have the variable in the denominator of one term. First, we subtract 6.25 from both sides to isolate the fraction on the left-hand side.

Now, we multiply both sides of the equation by 2x. This way, we will get rid of fractions.

Finally, we perform the required operations to isolate the variable.

We got x=-2 as the solution. However, since the original equation has the variable in the denominator, some values of x may not satisfy the equation. This is why we need to check whether it is a valid solution. We can do this by substituting -2 into the equation and verifying if a true statement is obtained.

A true statement was obtained. Therefore, we can be sure that x=-2 is the solution to the equation.

Consider the number line. What number should be in the circle?

Number line. Numbers -17 and 25 are marked. A circle is between -17 and 25. The distance from -17 to the circle is 4d and the distance from 25 to the circle is 10d.

From the graph, we can see that if we add 4d to -17, we will get the number in the circle. Number in the circle → -17 + 4d In a similar fashion, if we subtract 10d from 25 we will get the number in the circle. Number in the circle → 25 - 10d An equation in terms of d is formed by equating the previous two expressions. -17 + 4d = 25 - 10d Let's isolate the variable d by performing the required operations.

Now that we know the value of d, we can substitute it into either of the expressions to find the number in the circle. For example, let's substitute into - 17+4d.

The number in the circle is -5. Note that the substitution into 25-10d would give the same answer.

Izabella bought a cake and four bags of marshmallows. Altogether she paid

Let be the price of a bag of marshmallows. Write an equation that describes what Izabella has bought.
How much does a bag of marshmallows cost?

We know that the cake costs $25 and that Izabella bought only one. 1*Cake → $ 25 In addition, Izabella bought four bags of marshmallows. Since each bag costs x dollars, we can express the total cost of the marshmallows as 4x. 4*Bag of Marshmallows→ $ 4x In total, Izabella paid $ 61. This means that the sum of the two amounts is $61. Then, the following equation describes Izabella's purchase. 25 + 4x = 61

We need to find the cost of a bag of marshmallows. Since x represents this value, this means that we need to solve the equation 25+4x=61 for x. We will do this by applying inverse operations to isolate x on the left-hand side.

A bag of marshmallows costs $9.

Last weekend, Kevin read five-eighths of a book. He calculates that he read more pages than he still has to read.

Image of a book
How many pages does the book have?

Let T be the total number of pages in the book. Kevin has already read five-eighths of the book, which corresponds to 58T pages. This means that he still has to read three-eighths of the book, or 38T pages. Pages read &→ 5/8T [0.8em] Pages left &→ 3/8T According to Kevin's calculations, the number of pages left plus 120 is equal to the number of pages read. This information leads us to write the following equation. 3/8T + 120 = 5/8T To determine how many pages Kevin's book has, we will solve this equation for T. To start, we will multiply both sides of the equation by 8 to get rid of the fractions.

Now that we do not have fractions, we can continue solving the equation by using inverse operations.

From this, we can conclude that the Kevin's book has 480 pages.

A youth baseball team is holding a car wash fundraiser to raise money for new uniforms. To help with the fundraiser, they decided to rent a pressure washer for and charge to wash a car.

If renting the pressure washer is the only cost of the fundraiser, how many cars does the team have to wash to break even?
The cost of new uniforms for the entire team is How many cars does the team have to wash to cover the cost of the fundraiser and still afford new uniforms?

Let x be the number of cars washed. Since the team charges $3 for each car washed, the earnings from washing x cars is equal to 3x. Earnings → 3x From the given information, the team has to pay $51 to rent the pressure washer. This is the cost of running the fundraiser. Costs → 51 Breaking even means that the earnings minus the costs is equal to zero. Break even 3x - 51 = 0 The number of cars the team has to wash to break even is found by solving this equation for x.

The baseball team must wash 17 cars to break even.

In order to buy the new uniforms, the team has to raise $240. This means that after paying for the pressure washer, the team should still have $240 — that is, when we subtract 51 from 3x, the result is 240. 3x - 51 = 240 Let's solve this equation.

The team has to wash 97 cars in order to cover their costs and still be able to buy the new uniforms.

Solve each equation.

To solve an equation, we should first gather all of the variable terms on one side of the equation and all of the constant terms on the other by using the Properties of Equality. In this case, we have the variable on both sides, so we can start by adding 2t on both sides to group them on the left-hand side.

Now, we can continue solving the equation using the Properties of Equality.

The solution to the equation is t=8.


As in the previous part, we have the variable on both sides. We can start by collecting them on the right-hand side. To do so, we add g to both sides of the equation.

Next, we will collect the constant terms on the right-hand side.

Finally, we will divide both sides of the equation by -3. That way we will isolate the variable.

The solution to the equation is g=-4.

Tearrik bought a helicopter-shaped drone and went to the park to try it out. Once there, he placed the drone on its launch platform and turned it on. According to the instructions, when flying vertically upward, the drone's altitude in centimeters after seconds is given by the expression

A drone with the shape of a helicopter. The altitude after t seconds is given by the expression 25t+100
If Tearrik flies the drone vertically upwards, how long does it take for the drone to reach an altitude of centimeters?

Since Tearrik is flying the drone vertically upwards, the drone's altitude is given by 25t+100, where t is the number of seconds after launch. Drone's Altitude 25t + 100 We want to know how many seconds it takes for the drone to reach an altitude of 460 centimeters. To figure it out, we will set the previous expression equal to 460. 25t + 100 = 460 Let's solve this equation for t.

The solution is t=14.4. This means that it takes 14.4 seconds for the drone to reach an altitude of 460 centimeters.

Solving Multi-Step Equations in One Variable

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