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
To solve an equation, the variable needs to be isolated on one side. 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.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
Two 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 is accepted without proof. This property is one of the Properties of Equality that can be used when solving equations. Consider an example. x-3=5 Adding 3 to both sides of the equation will isolate the variable x and allow us to find the solution to the equation.
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 is accepted without proof. This property is one of the Properties of Equality that can be used when solving equations. Consider an example. x+2=7 Subtracting 2 from both sides of the equation will isolate the variable x and allow us to find the solution to the equation.
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Example
Getting Ready for the School Fair
Students are preparing for a school fair. They need to figure out several missing amounts, such as the number of supplies, tickets, and decorations.
Each missing amount is represented by a variable. Solve the equations to find the missing values.
a
p-5=3
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b
m+9=14
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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 using the Addition Property of Equality.
b
The variable m has to be isolated. Notice that 9 is added to m. To undo this addition, subtract 9 from both sides using the Subtraction Property of Equality.
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Pop Quiz
Solving Equations
Solve the equations 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 Ignacio's class, a certain number of people got sick and missed math class. There were 19 people present in class today. Ignacio'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 got sick and the number of people who were present is equal to the total number of people in the class. ⇓ x + 19 = 24 Use the Subtraction Property of Equality to solve for x, the number of students in Ignacio's class who got sick.
We found that 5 people in Ignacio's class got sick and missed class.
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Example
Snack Time
Ali is buying snacks and drinks. The snacks and a few sodas cost $ 13 in total. If the sodas cost $ 7, how much money did Ali spend 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 Ali spend on 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 and 7 is equal to 13. If x is the amount of money spent on the snacks, we can write the above sentence as the following equation. x + 7 = 13
b
We will isolate the x-variable on one side of the equation to solve the equation. Since 7 is added to x, let's 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. We found that x=6. That means the total cost of the snacks is $ 6.
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Example
Modeling Cycling Distance With an Equation
Davontay is riding his bike along a trail in a nature preserve.
He has already ridden 9 kilometers and has 3 kilometers left to reach the end of the trail. Let x represent the total length of the trail.
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 long is the trail?
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Hint
a
Assign a variable to the unknown quantity. In this case, the unknown quantity is the length of the trail.
b
Use the Addition Property of Equality.
c
What does the variable represent?
Solution
a
We will start by assigning a variable to the unknown quantity when writing an equation from a verbal description. Let x be the the length of the bike trail. Then, let's break down the given statement and write the equation.
The difference between the total length of the trail and the 9 kilomters Davontay has already ridden is equal to 3. ⇓ x - 9 = 3
b
Let's isolate the x-variable on one side of the equation using inverse operations. Since 9 is subtracted from x, we can use the Addition Property of Equality to do so.
c
Recall that x represents the total length of the bike trail. Since x=12, this means that the bike trail is 12 kilometers long.
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Closure
Calculating Diego's Age
Earlier we were told that Diego's younger brother is 12 years old and that the difference between their ages is 3 years.
External credits: @freepik
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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Hint
a
Assign a variable to Diego's age.
b
Use the Addition Property of Equality.
Solution
a
Let's start by assigning the variable x to the unknown quantity, Diego's age. Because Diego's brother is 12 years old, the difference between x and 12 is equal to 3. We will write the equation using this information.
x - 12 = 3
b
Let's isolate the x-variable using inverse operations to solve the equation. Since 12 is subtracted from x, we will use the Addition Property of Equality to do so.
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.
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Writing and Solving One-Step Addition and Subtraction Equations
Exercise 1.1
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We are asked to solve the given equation.
x+6=13
When solving equations, we can use inverse operations and Properties of Equality to undo
the operations applied to the variable. Here, 6 is added to the variable x.
x+ 6=13
To undo this operation, we use the inverse operation of addition — subtraction. The Subtraction Property of Equality lets us subtract 6 from both sides of the equation. Then, we will simplify.
The solution to our equation is x = 7.
We are asked to solve the given equation.
d+2=- 3
We can use inverse operations and Properties of Equality to undo
the operations applied to the variable. This lets us isolate the variable on one side and solve the equation. In this case, 2 is added to the variable d.
d+ 2=- 3
We will use the inverse operation of addition — subtraction. By the Subtraction Property of Equality, subtracting 2 from both sides of the equation results in an equivalent equation. This means that the resulting equation has the same solution as the original equation.
The solution to our equation is d = - 5.
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Solution
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{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":""},"formTextBefore":"g =","formTextAfter":null,"answer":{"text":["13"]}}
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We need to solve the given equation.
x-2=7
When solving equations, we can use inverse operations and Properties of Equality to undo
the operations applied to the variable. Here, 2 is subtracted from the variable x.
x- 2=7
We use the inverse operation of subtraction to undo this operation. That would be the operation of addition. We can add 2 to both sides of the equation by the Addition Property of Equality. Let's do it! Then, we will simplify.
The solution to our equation is x = 9.
We can start by considering the given equation.
g-5=8
Let's use inverse operations and Properties of Equality to undo
the operations applied to the variable g. We can see that 5 is subtracted from g in this equation.
g- 5= 8
Like in Part A, let's use the inverse operation of subtraction to undo this operation. Again, that would be the operation of addition. The Addition Property of Equality lets us add 5 to both sides of the equation. Then, we will simplify.
The solution to our equation is g = 13.
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Solution
{"type":"multichoice","form":{"alts":["a - 9 = 4","a + 2 = 15","a - 8 = 4","a - 6 = 2"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":[0,1]}
{"type":"multichoice","form":{"alts":["x + 3 = 2","x + 2= 10","x - 4 = 4","x + 5 = - 2"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":[1,2]}
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We are asked to choose all the equations that are equivalent to the given equation a - 8 = 5. Two equations are equivalent if they have the same solutions. a - 9 &= 4 a + 2 &= 15 a - 8 &= 4 a - 6 &= 2 By the Addition and Subtraction Properties of Equality, adding or subtracting the same number from both sides of an equation results in an equivalent equation. We can use these properties to show that two equations are equivalent by transforming one into the other.
Equation I: a - 9 = 4
Consider the equation a-9=4. There is a - 9 on the left-hand side. This expression is equivalent to a - 8 - 1. Let's take the given equation a-8=5 and subtract 1 from each side. In other words, we will apply the Subtraction Property of Equality.
Subtracting 1 from each side of the equation a - 8 = 1 results in the equation a - 9 = 4. This means that a - 8 = 5 and a - 9 = 4 are equivalent equations.
Equation II: a +2 = 15
Next, we will consider the equation a + 2 = 15. Note that a + 2 is equivalent to a - 8 + 10. Then, we can take the given equation a - 8 = 5 and add 10 to each side. Let's do it!
This action results in the equation a + 2 = 15. Therefore a-8 = 5 and a + 2 = 15 are equivalent equations by the Addition Property of Equality.
Equation III: a-8=4
Let's consider the next equation, a - 8 = 4. It is not possible for a - 8 to be equal to 5 and 4 at the same time. Given:& a-8= 5 Considered:& a-8= 4 This means that these equations have different solutions. Therefore, they are not equivalent equations.
Equation IV: a - 6 = 2
Finally, let's consider the equation a - 6 = 2. The left-hand side a - 6 is equivalent to a - 8 + 2. Let's add 2 to each side of the equation a - 8 = 5. We want to check if we will get the equation we are considering.
The equations a - 8 = 5 and a - 6 = 7 are equivalent. That is the case because of the Addition Property of Equality. However, there is a 2 instead of a 7 on the right-hand side of our equation. a - 6 = 2 a - 6 = 7 These two equations have different solutions. This means that they are not equivalent equations. Therefore, a - 6 = 2 and a - 8 = 5 are not equivalent equations either.
Conclusion
We found that two of the given equations, a-9=4 and a+2=15, are equivalent to the equation a-8=5.
We are asked to choose all the equations that are equivalent to the given equation x - 1 = 7.
x + 3 &= 2
x+2 &= 10
x-4 &= 4
x+5 &= - 2
Remember that two equations are equivalent when they have the same solutions. We can check if two equations are equivalent by solving them and comparing their solutions. First, let's solve the given equation.
x - 1 = 7
Here, 1 is subtracted from the variable x. We isolate the variable on one side of the equation by undoing
the operations applied to the variable using inverse operations. The inverse operation of subtraction is addition, so we use the Addition Property of Equality to add 1 to both sides of the equation.
The solution to the given equation is x = 8. Now, let's solve the remaining equations, starting with x + 3 = 2. Here, 3 is added to the variable. Then, we will use the Subtraction Property of Equality to subtract 3 from both sides of the equation.
We use the same method to solve the remaining equations.
| Equation | Solution |
|---|---|
| x+3=2 | x = - 1 |
| x + 2 = 10 | x = 8 |
| x -4 = 4 | x = 8 |
| x + 5 = - 2 | x = - 7 |
We can see that two equations have the same solution as the given equation: x + 2 = 10 and x - 4 = 4. This means that only these two equations are equivalent to the given equation.
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{"type":"multichoice","form":{"alts":["x + 6 = 14","x + 2 = 11","x + 1 = 9","x + 3 = 12"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":[0,2]}
{"type":"multichoice","form":{"alts":["d + 3 = 0","d + 10= 11","d + 7 = 0","d + 1 = - 6"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":[2,3]}
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We are asked to choose all the equations that are equivalent to the given equation x + 2 = 10. x + 6 &= 14 x + 2 &= 11 x + 1 &= 9 x + 3 &= 12 Two equations are equivalent if they have the same solutions. By the Addition and Subtraction Properties of Equality, adding or subtracting the same number from both sides of an equation results in an equivalent equation. We can use this to show that two equations are equivalent by transforming one into the other.
Equation I: x + 6 = 14
Consider the equation x+6=14. There is x + 6 on the left-hand side. Note that this is equivalent to x + 2 +4. Let's take the given equation x+2=10 and add 4 to each side. In other words, we will apply the Addition Property of Equality.
Adding 4 to each side of the equation x + 2 = 10 results in the equation x + 6 = 14. This means that the equations x + 2 = 10 and x + 6 = 14 are equivalent equations.
Equation II: x+2=11
Let's consider the next equation, x + 2 = 11. It is not possible for x + 2 to be equal to 11 and 10 at the same time. Given:& x+2= 10 Considered:& x+2= 11 This means that these equations have different solutions. Therefore, they are not equivalent equations.
Equation III: x +1 = 9
Next, we will consider the equation x + 1 = 9. Note that x + 1 is equal to x + 2 - 1. Then, we can take the given equation x + 2 = 10 and subtract 1 from each side. Let's do it!
This action results in the equation x + 1 = 9. Therefore x+2 = 10 and x + 1 = 9 are equivalent equations by the Subtraction Property of Equality.
Equation IV: x+3=12
Finally, let's consider the equation x + 3 = 12. The left-hand side x + 3 is equal to x + 2 + 1. Let's add 1 to each side of the equation x + 2 = 10. We want to check if we will get the equation we are considering.
Thanks to the Addition Property of Equality, we can note that the equations x +2 = 10 and x + 3 = 11 are equivalent. Wait a minute! It is 12 and not 11 on the right-hand side of our equation. This makes things a bit more fun. Let's take a closer look. x + 3 = 12 x + 3 = 11 These two equations have different solutions. This means that they are not equivalent equations. Therefore, x+3=12 and x+2=10 are not equivalent equations either.
Conclusion
We found that two of the given equations, x+6=14 and x+1=9, are equivalent to the equation x+2=10.
We are asked to choose all the equations that are equivalent to the given equation d + 8 = 1.
d + 3 &= 0
d+10 &= 11
d+7 &= 0
d+1 &= - 6
Two equations are equivalent when they have the same solutions. We can check if two equations are equivalent by solving them and comparing their solutions. Let's start by solving the given equation.
d + 8 = 1
Here, 8 is added to the variable d. We can isolate the variable on one side by undoing
the operations applied to the variable using inverse operations. The inverse operation of addition is subtraction, so we will subtract 8 from both sides of the equation using the Subtraction Property of Equality.
The solution to the given equation is d = - 7. Now, let's solve each of the remaining equations. Remember, when a number is added to the variable, we use the Subtraction Property of Equality to isolate the variable on one side.
| Equation | Apply Properties of Equality | Solution |
|---|---|---|
| d+3 = 0 | d + 3 - 3 = 0 - 3 | d = - 3 |
| d + 10 = 11 | d + 10 - 10 = 11 - 10 | d = 1 |
| d + 7 = 0 | d + 7 - 7 = 0 - 7 | d = - 7 |
| d + 1 = - 6 | d + 1 - 1 = - 6 - 1 | d = - 7 |
The equations d + 7 = 0 and d + 1 = - 6 have the same solution as the given equation. This means that these two equations are equivalent to the given equation. The remaining equations have different solutions, so they are not equivalent to the given equation.
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Writing and Solving One-Step Addition and Subtraction Equations
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