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Almost all real-life situations can be represented by equations, which makes them very important in math. It is possible to use equations to model real-life situations and predict their outcomes. This lesson will discuss how to represent real-life situations using equations, and how to solve these equations to find a solution.

Catch-Up and Review

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

Equation
Variable
Integer Numbers

Challenge

How Old Is Diego?

Diego and his younger brother dream of playing college football together on the same team. Right now his younger brother is $12$ years old. There is a $3$ year difference between their ages.

a Write an equation in terms of

x

that represents the situation.

b Solve the equation to find Diego's age.

Discussion

Equivalent Equations

The variable needs to be isolated on one side of an equation in order to solve the equation. This can be achieved by undoing certain operations using inverse operations.

Concept

Inverse Operations

Inverse operations are two operations that undo one another. For example, adding

6

and subtracting

6

are inverse operations because they cancel each other out. This means that adding

6

to any number and then subtracting

6

results in the original number.

10 + 6 - 6 10 + 6 - 6 10

Equations are solved by using inverse operations. By the Properties of Equality, any operation performed on one side of an equation must also be performed on the other side of the equation to maintain equality. Consider an example.

x - 4 = 9

This equation can be solved by adding

4

to both sides.

x - 4 = 9

AddEqn

$LHS + 4 = RHS + 4$

x - 4 + 4 = 9 + 4

Simplify left-hand side

x - 4 + 4 = 9 + 4

AddTerms

Add terms

x = 13

In this case, the subtraction on the left-hand side of the equation can only be eliminated by adding

4

on both sides. The result of applying the Properties of Equality on an equation is an equivalent equation.

Concept

Equivalent Equation

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

Here, two equations that are equivalent to the given equation were created as a result of applying the Subtraction Property of Equality. They are equivalent because

15

is the solution to all these equations.

I . II . III . y + 3 = 18 y + 3 - 3 = 18 - 3 y = 15

The obtained solution can be checked by substituting

y = 15

in the original equation. If a true statement results, the solution is correct.

y + 3 = 18

Substitute

$y = 15$

15 + 3 = ? 18

AddTerms

Add terms

18 = 18 ✓

A true statement was obtained. Therefore,

y = 15

is indeed a solution to the original equation.

Discussion

Addition and Subtraction Properties of Equality

Some of the most commonly used inverse operations are addition and subtraction. These operations fall under the Addition Property of Equality and the Subtraction Property of Equality.

Rule

Addition Property of Equality

Adding the same number to both sides of an equation results in an equivalent equation. Let $a,$ $b,$ and $c$ be real numbers.

If $a = b,$ then $a + c = b + c .$

The Addition 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 an example.

x - 3 = 5

By adding

3

to both sides of the equation, the variable

x

can be isolated and the solution to the equation can be found.

x - 3 = 5

AddEqn

$LHS + 3 = RHS + 3$

x - 3 + 3 = 5 + 3

Simplify left-hand side

x - 3 + 3 = 5 + 3

AddTerms

Add terms

x = 8

Rule

Subtraction Property of Equality

Subtracting the same number from both sides of an equation results in an equivalent equation. Let $a,$ $b,$ and $c$ be real numbers.

If $a = b,$ then $a - c = b - c .$

The Subtraction 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 an example.

x + 2 = 7

By subtracting

2

from both sides of the equation, the variable

x

can be isolated and the solution to the equation can be found.

x + 2 = 7

SubEqn

$LHS - 2 = RHS - 2$

x + 2 - 2 = 7 - 2

Simplify left-hand side

x + 2 - 2 = 7 - 2

SubTerms

Subtract terms

x = 5

Example

Doing the Dishes Together

Diego and his younger brother continued to talk about their college football dreams. Their abuelo — grandpa — overheard their dream and told them a little secret. He was a college football player! More importantly, he said he had to wash dishes to help pay for college. Diego's abuelo showed Diego a photo of him doing dishes at home after his playing days were long over. Diego becomes more curious about washing dishes than any football dreams. He asks his abuelo two questions about the night the picture was taken.

How many plates were washed?
How long did it take to finish the dishes?

Diego's abuelo really wants to help Diego with math. He writes two equations whose solutions are the answers to Diego's questions. Find the answers to the questions by solving the equations.

p - 5 = 3

m + 9 = 14

Hint

a Use the Addition Property of Equality.

b Use the Subtraction Property of Equality.

Solution

a The variable

p

must be isolated to solve the equation. On the left-hand side

5

is subtracted from

p .

This subtraction can be undone by adding

5

to both sides by the Addition Property of Equality.

p - 5 = 3

AddEqn

$LHS + 5 = RHS + 5$

p - 5 + 5 = 3 + 5

Simplify left-hand side

p - 5 + 5 = 3 + 5

AddTerms

Add terms

p = 8

That night,

8

plates were washed.

b This time, the variable

m

has to be isolated. Notice that

9

is added to

m .

To undo this addition, subtract

9

from both sides by using the Subtraction Property of Equality.

m + 9 = 14

SubEqn

$LHS - 9 = RHS - 9$

m + 9 - 9 = 14 - 9

Simplify left-hand side

m + 9 - 9 = 14 - 9

SubTerms

Subtract terms

m = 5

It took

5

minutes to do the dishes that night. Not bad at all! Diego realizes that he should help wash dishes more often at home, too.

Pop Quiz

Solving Equations

Solve the equations by using the Addition Property of Equality or the Subtraction Property of Equality.

Discussion

Writing Equations

Some real-life situations can be algebraically modeled by equations. A critical step in doing this is to represent an unknown quantity with a variable. Consider the following situation.

In Diego's class, a certain number of people became sick and missed math class. There were 19 people present in class, and Diego's class has 24 people in total.

Here, the unknown quantity is the number of people who became sick. It can be represented by the variable

x .

In order to write an equation, a verbal statement needs to be translated into an algebraic expression including a number and an equals sign.

The sum of the number of people who fell sick and the number of people who were present is equal to the total number of people in the class . ⇓ x + 19 = 24

The equation can now be solved to find the number

x

of people in Diego's class who fell sick. Use the Subtraction Property of Equality.

x + 19 = 24

SubEqn

$LHS - 19 = RHS - 19$

x + 19 - 19 = 24 - 19

Simplify left-hand side

x + 19 - 19 = 24 - 19

SubTerms

Subtract terms

x = 5

Recall that

x

represents the number of people who became sick. This means it can be concluded that

5

people in Diego's class became sick.

Example

Connecting Math to the Real World

Diego's abuleo gets a great deal of delight from seeing Diego so interested in math.

However, he realized that Diego is having some issues with connecting math to the real world. For this reason, he told Diego that he will buy some snacks and sodas to share if Diego can answer the following question.

Some snacks and a few sodas cost, in total, $$ 13 .$ If the sodas cost $$ 7,$ how much money is spent on snacks?

a Write an equation in terms of

x

to represent the situation.

b Find the value of

x

for the equation written in Part A.

c How much money did Diego's abuelo spend on the snacks?

Hint

a Assign a variable to the unknown quantity. In this case, the unknown quantity is the cost of the snacks.

b Use the Subtraction Property of Equality.

c What does the variable represent?

Solution

a The first thing to do when writing an equation from a verbal description is to assign a variable to the unknown quantity. The given statement is equivalent to the following sentence.

The sum of the money spent on snacks and 7 is equal to 13 .

x

is the amount of money spent on the snacks, the equation can be written by following this statement.

x + 7 = 13

b Isolate the

x -

variable on one side of the equation to solve the equation. Since

7

is added to

x,

use the Subtraction Property of Equality.

x + 7 = 13

SubEqn

$LHS - 7 = RHS - 7$

x + 7 - 7 = 13 - 7

Simplify left-hand side

x + 7 - 7 = 13 - 7

SubTerms

Subtract terms

x = 6

c Remember that

x

represents the amount of money spent on the snacks. The calculations show that

x = 6 .

That means the total cost of the snacks is

$ 6 .

It looks like Diego is going to enjoy some snacks and soda!

x = 6 \Rightarrow The price of the snacks is $ 6 .

Example

Modelling Cycling Distance With an Equation

Diego's abuelo remembered Diego's goal to become a college football player. He felt so guilty about giving Diego so much junk food! He thinks he should teach Diego about a healthy and active lifestyle. He tells Diego about how he rode his bicycle everyday when he was young. Again, Diego's abuelo wanted to give his grandson a math problem about his own history. Diego, when I was young I rode so much you wouldn't believe it. In fact, the difference between the number of kilometers I used to ride and $9$ is equal to $3 .$

a Write an equation in terms of

x

to represent the situation.

b Find the value of

x

for the equation written in Part A.

c How many kilometers did Diego's abuelo cycle every day?

Hint

a Assign a variable to the unknown quantity. In this case, the unknown quantity is the number of kilometers Diego's granddad used to cycle per day.

b Use the Addition Property of Equality.

c What does the variable represent?

Solution

a Begin by assigning a variable to the unknown quantity when writing an equation from a verbal description. Let

x

be the number of kilometers that Diego's abuelo cycled. Then, breakdown the given statement.

The difference between the number of kilometers I used to bike and 9 is equal to 3 . ⇓ x - 9 = 3

b Isolate the

x -

variable on one side of the equation to solve it. Notice that

9

is subtracted from

x .

That means the Addition Property of Equality can be used.

x - 9 = 3

AddEqn

$LHS + 9 = RHS + 9$

x - 9 + 9 = 3 + 9

Simplify left-hand side

x - 9 + 9 = 3 + 9

AddTerms

Add terms

x = 12

c Recall that

x

represents the number of kilometers that Diego's abuelo cycled every day. Again, note what

x

equals. Since

x = 12,

this means that Diego's abuelo cycled

12

kilometers every day.

x = 12 ⇓ He cycled 12 kilometers every day .

Closure

Calculating Diego's Age

The challenge presented at the beginning of the lesson can be solved by applying the learned concepts. Recall that the challenge stated that Diego's younger brother is

12

years old and that the difference between Diego's age and his brother's age is

3

years.

External credits: @freepik

The two brothers continued to dream about becoming college football players. Diego then felt inspired by all of his abuelo's hard work. This drove Diego to figure out how to write the equation that determines his age. After all, if he can not do math, how can he succeed as an athlete? He thought.

a Write an equation in terms of

x

that represents the situation.

b Solve the equation to find Diego's age.

Hint

a Assign a variable to Diego's age.

b Use the Addition Property of Equality.

Solution

a The first thing to do is to assign a variable to the unknown quantity. In this case, the unknown quantity is Diego's age, so let it be

x .

Because Diego's brother is

12

years old,

the

difference

between

x

and

12

is

equal

to

3 .

With this information, the equation can be written.

x - 12 = 3

Diego figured out the equation!

b The

x -

variable must be isolated to solve the equation. In this case,

12

is subtracted from

x .

That means it is possible to use the Addition Property of Equality to isolate

x .

x - 12 = 3

AddEqn

$LHS + 12 = RHS + 12$

x - 12 + 12 = 3 + 12

Simplify left-hand side

x - 12 + 12 = 3 + 12

AddTerms

Add terms

x = 15

Diego is

15

years old. He already knew that, but finding the solution made him feel great knowing that he also wrote the equation correctly. Diego's abuelo is so proud!

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Catch-Up and Review

Hint

Solution

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Solution

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Solution

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Solution