1. Writing and Solving One-Step Addition and Subtraction Equations
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Chapter 6
1.
Writing and Solving One-Step Addition and Subtraction Equations
This lesson aims to teach the fundamentals of solving one-step equations, specifically focusing on addition and subtraction methods. It emphasizes the importance of understanding properties of equality to solve these equations effectively. Real-world scenarios are integrated into the teaching approach, making the learning experience more relatable. For example, the material uses everyday situations like calculating the cost of snacks or determining the number of people who fell sick in a class to illustrate the practical applications of one-step equations. By grasping these basic concepts, learners can tackle a wide range of problems, from personal finance to health statistics, thereby making informed decisions in daily life.
Show less Show more expand_more| Student Learning Objectives: |
|---|
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| | 10 Theory slides |
| | 9 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Writing and Solving One-Step Addition and Subtraction Equations
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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.
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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.
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b
Solve the equation to find Diego's age.
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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.
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
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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.
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.
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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.
a
p-5=3
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b
m+9=14
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":"m=","formTextAfter":null,"answer":{"text":["5"]}}
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.
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.
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Pop Quiz
Solving Equations
Solve the equations by using the Addition Property of Equality or the Subtraction Property of Equality.
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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.
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.
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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.
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b
Find the value of x for the equation written in Part A.
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c
How much money did Diego's abuelo spend on the snacks?
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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 and7 is equal to 13. If 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 ⇒ The price of the snacks is$ 6.
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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.
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b
Find the value of x for the equation written in Part A.
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c
How many kilometers did Diego's abuelo cycle every day?
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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.
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 12kilometers every day.
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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
a
Write an equation in terms of x that represents the situation.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["x-12=3","3=x-12","x-3=12","12=x-3","x=3+12","x=12+3"]}}
b
Solve the equation to find Diego's age.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":""},"formTextBefore":"Diego is","formTextAfter":"years old.","answer":{"text":["15"]}}
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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Writing and Solving One-Step Addition and Subtraction Equations
Exercise 2.1
Heichi is reading a book called the Game of Roses. It is so thrilling!
He already read 107 pages in just a few days. The book is 314 pages long. Write an equation in terms of x that can be used to find the number of pages that Heichi has left to read.
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We want to write an equation for the number of pages Heichi still needs to read. An unknown quantity is represented by a variable in an equation. In our case, the unknown quantity is the number of remaining pages. Let's represent it by the variable x. Number of Remaining Pages: x We know that Heichi has read 107 pages of a book that is 314 pages long. The total number of pages is the sum of the number of pages Heichi read and the number of remaining pages. Sum of 107 and x is equal to 314. ⇓ 107 + x = 314 The equation 107 + x = 314 can be used to find the number of pages remaining in the book. Note that this is only one possible way of writing an equation describing this situation.
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Solution
Ramsha is huffing and puffing her way towards the finish of a bike race with her friends.
The race is 7 miles long and she is 3 miles from the finish line. Write an equation in terms of d that can be used to find the distance Ramsha has cycled so far.
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We want to write an equation that can be used to find the distance Ramsha has cycled so far. In an equation, the variable represents some unknown quantity. In our case, the unknown quantity is the distance Ramsha has cycled. We are told to use d as the variable. Distance Cycled: d We know that Ramsha is 3 miles away from the finish line, and she has cycled d miles. The total distance of the race, 7 miles, is equal to the sum of the distance Ramsha has cycled and the remaining distance to the finish line. Sum of d and 3 is equal to 7. ⇓ d + 3 = 7 The equation d + 3 = 7 can be used to find the distance Ramsha has cycled.
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Consider the following equation. x - 2 = 8 Choose the situation that can be modeled by the given equation. A. & Tearrik has a certain number of apples. He gives two to & his friend. He has eight apples left. [0.4em] B. & Tearrik has a certain number of apples. His friend gives & him two more. He now has eight apples in total. [0.4em] C. & Tearrik has a certain number of apples. He gives eight & to his friend. He has two apples left. [0.4em] D. & Tearrik has a certain number of apples. His friend gives &him eight more. He now has two apples in total.
{"type":"choice","form":{"alts":["A","B","C","D"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":0}
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We are given an equation. x - 2 = 8 Here, 2 is subtracted from the variable x. The result of this subtraction is 8. The variable represents some unknown quantity in an equation. In each of the given situations, the unknown quantity is the original number of apples Tearrik had. The variable x represents this quantity. Number of Apples: x Then, 2 is subtracted from x, so the number of apples decreases by 2. This can mean that Tearrik gives two apples to a friend. If his friend gave him some apples, the number x of apples increases. The result of this subtraction is 8. This is how many apples are left. x - 2 = 8 [0.3em] Tearrik has xapples and gives 2to his friend. He has 8apples left. This situation is described in option A.
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Consider the following equation. r + 4 = 17 Choose the situation that can be modeled by this equation. A. & On Monday, a number of runners signed up & for a race. On Tuesday,4 runners backed out. & In total,17runners are taking part in the race. [0.4em] B. & On Monday, a number of runners signed up for & a race. On Tuesday,17more runners signed up. & In total,4runners are taking part in the race. [0.4em] C. & On Monday, a number of runners signed up for & a race. On Tuesday,17runners backed out. & In total,4runners are taking part in the race. [0.4em] D. & On Monday, a number of runners signed up for & a race. On Tuesday,4more runners signed up. & In total,17runners are taking part in the race.
{"type":"choice","form":{"alts":["A","B","C","D"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":3}
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Consider the given equation. r + 4 = 17 In our equation, 4 is added to the variable r. The result of this addition is 17. The variable represents some unknown quantity in an equation. In each of the given situations, the unknown quantity is the number of runners who signed up for the race on Monday. The variable r represents this quantity. Runners Signed Up On Monday: r Then, 4 is added to r, so the number of runners increases by 4. This can mean that 4 more runners signed up on Tuesday. If some runners backed out, the number r of runners would decrease. The result of this addition is 17, so the total number of runners is 17. r + 4 = 17 [0.5em] On Monday, rrunners signed up. On Tuesday, 4 more runners signed up. In total, 17runners are taking part. This situation is described by option D.
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Writing and Solving One-Step Addition and Subtraction Equations
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