Lesson Objectives

- Demonstrate an understanding of Algebraic Expressions
- Demonstrate an understanding of the Additive Inverse Property
- Demonstrate an understanding of the Additive Identity Property
- Learn about the Addition Property of Equality
- Learn how to solve an equation using Addition
- Learn how to solve an equation using Subtraction

## How to Solve Equations using Addition and Subtraction

In the last lesson, we learned that an equation is a
statement that two algebraic expressions are equal. We also learned that
a solution to an equation is any value that replaces the variable and makes the equation true. Again, we say "true" to mean the
left side of the equation and the right side of the equation are the same value or are "equal". As an example, suppose we are asked if
x = 3 is a solution to the equation:
2x - 1 = 5

We can check by plugging in a 3 for x and evaluating.

2x - 1 = 5

2(3) - 1 = 5

6 - 1 = 5

5 = 5

Since the value on the left and right are the same (5), we know the solution (x = 3) is correct.

As another example, suppose we asked if x = -4 is a solution to the equation:

12x - 5 = 20

We can check by plugging in a (-4) for x and evaluating.

12x - 5 = 20

12(-4) - 5 = 20

-48 - 5 = 20

-53 = 20 (false!)

Since the value on the left (-53) is not the same as the value on the right (20), we know x = -4 is not the correct solution.

Before we attempt a problem, we need to learn about the addition property of equality. The Addition Property of Equality tells us that we can add or subtract any value to or from both sides of an equation without changing the solution. Additionally, we need to think about two additional properties that we learned in pre-algebra. First, recall the additive inverse property. This property states that any number plus its opposite results in 0:

5 + (-5) = 0

17 + (-17) = 0

Second, recall that zero is the additive identity. This means that any number plus 0 leaves the number unchanged:

131 + 0 = 131

-25 + 0 = -25

These properties may seem obvious when shown in this format, but they may not be so obvious when solving equations. Let's put these three properties together and show how to solve an equation using addition or subtraction. Let's suppose we encountered a simple equation such as:

x - 2 = 7

Most of us can inspect the equation and determine that x = 9 is the solution since 9 - 2 = 7. While this approach works here, we will face more difficult equations and need a step by step process. Our goal when solving an equation is to isolate our variable on one side of the equation. In order to do that, we use inverse operations to produce equivalent equations (equations with the same solution(s)) until we have the format of:

x = some number

In this case, when we say "inverse operations", we mean to think of addition and subtraction as inverse operations. This means one undoes the other. This goes back to our rule about adding opposites results in 0. If we have 6 + 5 and subtract away 5, we are undoing the + 5 part:

6 + 5 - 5 = 6 + 0 = 6

In the case of our equation, x - 2 = 7, we want to think about how to isolate x.

In our case, we can think about x - 2 = 7 as x + (-2) = 7. We want to get rid of the (-2) that's being added to x so that x is by itself on the left side of the equation. (-2) + 2 = 0, so we can add 2 to both sides of the equation. This will get rid of the + (-2) on the left side and leave x by itself. Another way to think about this is x has a 2 that is being subtracted away, we can isolate x by adding 2 to each side of the equation:

x - 2 + 2 = 7 + 2

If we simplify each side:

left side » x - 2 + 2 = x + 0 = x

right side » 7 + 2 = 9

Putting the two sides together:

x = 9

In some cases, the steps viewed through a series of images may help: We start by stating that x - 2 is equal to or the same as 7. We add 2 to the left side of the equation to isolate x, but to make this legal, we must also add 2 to the right side. Once we simplify each side, we see that x is equal to 9.

It's okay if you don't completely understand at first. Solving equations may take quite a bit of practice to get the hang of the process. Let's take a look at a few examples.

Example 1: Solve each equation:

x + 3 = 10

We want to isolate x on the left side of the equation. Think about what is being done to x. In this case, 3 is being added to x. What can we add to 3 to make it go away?

3 + (-3) = 0

We could also think about this with straight subtraction, either works the same:

3 - 3 = 0

We can add (-3) to both sides of the equation or we can subtract 3 away from each side of the equation. It's the exact same either way, let's subtract 3 away from each side:

x + 3 - 3 = 10 - 3

Now we can simplify each side:

left side » x + 3 - 3 = x + 0 = x

right side » 10 - 3 = 7

Putting the two sides together:

x = 7

We can check our answer by plugging a 7 in for x in our original equation.

x + 3 = 10

7 + 3 = 10

10 = 10

Since the left and right side are the same value (10), we know our solution x = 7 is correct.

Example 2: Solve each equation:

x - 12 = 4

We want to isolate x on the left side of the equation. Think about what is being done to x. In this case, 12 is being subtracted away from x. What can we add to make - 12 go away? We can think about this as: x + (-12) = 4

(-12) + 12 = 0

Let's add 12 to each side of the equation:

x - 12 + 12 = 4 + 12

Now we can simplify each side:

left side » x - 12 + 12 = x + 0 = x

right side » 4 + 12 = 16

Putting the two sides together:

x = 16

We can check our answer by plugging a 16 in for x in our original equation.

x - 12 = 4

16 - 12 = 4

4 = 4

Since the left and right side are the same value (4), we know our solution x = 16 is correct.

Example 3: Solve each equation:

x + 13 = -18

We want to isolate x on the left side of the equation. Think about what is being done to x. In this case, 13 is being added to x. What can we add to make 13 go away? We can add (-13) to each side or we can subtract 13 away from each side.

13 + (-13) = 0

Let's subtract 13 away from each side of the equation:

x + 13 - 13 = -18 - 13

Now we can simplify each side:

left side » x + 13 - 13 = x + 0 = x

right side » -18 - 13 = -31

Putting the two sides together:

x = -31

We can check our answer by plugging a -31 in for x in our original equation.

x + 13 = -18

-31 + 13 = -18

-18 = -18

Since the left and right side are the same value (-18), we know our solution x = -31 is correct.

We can check by plugging in a 3 for x and evaluating.

2x - 1 = 5

2(3) - 1 = 5

6 - 1 = 5

5 = 5

Since the value on the left and right are the same (5), we know the solution (x = 3) is correct.

As another example, suppose we asked if x = -4 is a solution to the equation:

12x - 5 = 20

We can check by plugging in a (-4) for x and evaluating.

12x - 5 = 20

12(-4) - 5 = 20

-48 - 5 = 20

-53 = 20 (false!)

Since the value on the left (-53) is not the same as the value on the right (20), we know x = -4 is not the correct solution.

### Addition Property of Equality

Now that we know how to check a given solution, how can we solve an equation? Solving an equation can seem very challenging at first, but if we follow a step by step process, it becomes very easy over time. Let's begin with the simplest type of equation. This is an equation which can be solved in one step using addition or subtraction.Before we attempt a problem, we need to learn about the addition property of equality. The Addition Property of Equality tells us that we can add or subtract any value to or from both sides of an equation without changing the solution. Additionally, we need to think about two additional properties that we learned in pre-algebra. First, recall the additive inverse property. This property states that any number plus its opposite results in 0:

5 + (-5) = 0

17 + (-17) = 0

Second, recall that zero is the additive identity. This means that any number plus 0 leaves the number unchanged:

131 + 0 = 131

-25 + 0 = -25

These properties may seem obvious when shown in this format, but they may not be so obvious when solving equations. Let's put these three properties together and show how to solve an equation using addition or subtraction. Let's suppose we encountered a simple equation such as:

x - 2 = 7

Most of us can inspect the equation and determine that x = 9 is the solution since 9 - 2 = 7. While this approach works here, we will face more difficult equations and need a step by step process. Our goal when solving an equation is to isolate our variable on one side of the equation. In order to do that, we use inverse operations to produce equivalent equations (equations with the same solution(s)) until we have the format of:

x = some number

In this case, when we say "inverse operations", we mean to think of addition and subtraction as inverse operations. This means one undoes the other. This goes back to our rule about adding opposites results in 0. If we have 6 + 5 and subtract away 5, we are undoing the + 5 part:

6 + 5 - 5 = 6 + 0 = 6

In the case of our equation, x - 2 = 7, we want to think about how to isolate x.

In our case, we can think about x - 2 = 7 as x + (-2) = 7. We want to get rid of the (-2) that's being added to x so that x is by itself on the left side of the equation. (-2) + 2 = 0, so we can add 2 to both sides of the equation. This will get rid of the + (-2) on the left side and leave x by itself. Another way to think about this is x has a 2 that is being subtracted away, we can isolate x by adding 2 to each side of the equation:

x - 2 + 2 = 7 + 2

If we simplify each side:

left side » x - 2 + 2 = x + 0 = x

right side » 7 + 2 = 9

Putting the two sides together:

x = 9

In some cases, the steps viewed through a series of images may help: We start by stating that x - 2 is equal to or the same as 7. We add 2 to the left side of the equation to isolate x, but to make this legal, we must also add 2 to the right side. Once we simplify each side, we see that x is equal to 9.

It's okay if you don't completely understand at first. Solving equations may take quite a bit of practice to get the hang of the process. Let's take a look at a few examples.

Example 1: Solve each equation:

x + 3 = 10

We want to isolate x on the left side of the equation. Think about what is being done to x. In this case, 3 is being added to x. What can we add to 3 to make it go away?

3 + (-3) = 0

We could also think about this with straight subtraction, either works the same:

3 - 3 = 0

We can add (-3) to both sides of the equation or we can subtract 3 away from each side of the equation. It's the exact same either way, let's subtract 3 away from each side:

x + 3 - 3 = 10 - 3

Now we can simplify each side:

left side » x + 3 - 3 = x + 0 = x

right side » 10 - 3 = 7

Putting the two sides together:

x = 7

We can check our answer by plugging a 7 in for x in our original equation.

x + 3 = 10

7 + 3 = 10

10 = 10

Since the left and right side are the same value (10), we know our solution x = 7 is correct.

Example 2: Solve each equation:

x - 12 = 4

We want to isolate x on the left side of the equation. Think about what is being done to x. In this case, 12 is being subtracted away from x. What can we add to make - 12 go away? We can think about this as: x + (-12) = 4

(-12) + 12 = 0

Let's add 12 to each side of the equation:

x - 12 + 12 = 4 + 12

Now we can simplify each side:

left side » x - 12 + 12 = x + 0 = x

right side » 4 + 12 = 16

Putting the two sides together:

x = 16

We can check our answer by plugging a 16 in for x in our original equation.

x - 12 = 4

16 - 12 = 4

4 = 4

Since the left and right side are the same value (4), we know our solution x = 16 is correct.

Example 3: Solve each equation:

x + 13 = -18

We want to isolate x on the left side of the equation. Think about what is being done to x. In this case, 13 is being added to x. What can we add to make 13 go away? We can add (-13) to each side or we can subtract 13 away from each side.

13 + (-13) = 0

Let's subtract 13 away from each side of the equation:

x + 13 - 13 = -18 - 13

Now we can simplify each side:

left side » x + 13 - 13 = x + 0 = x

right side » -18 - 13 = -31

Putting the two sides together:

x = -31

We can check our answer by plugging a -31 in for x in our original equation.

x + 13 = -18

-31 + 13 = -18

-18 = -18

Since the left and right side are the same value (-18), we know our solution x = -31 is correct.

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