# The Language of Algebra

## Solving Equations Using the Subtraction and Addition Properties of Equality

When some people hear the word algebra, they think of solving equations. The applications of solving equations are limitless and extend to all careers and fields. In this section, we will begin solving equations. We will start by solving basic equations, and then as we proceed through the course we will build up our skills to cover many different forms of equations.

### Determine Whether a Number is a Solution of an Equation

Solving an equation is like discovering the answer to a puzzle. An algebraic equation states that two algebraic expressions are equal. To solve an equation is to determine the values of the variable that make the equation a true statement. Any number that makes the equation true is called a solution of the equation. It is the answer to the puzzle!
To find the solution to an equation means to find the value of the variable that makes the equation true. Can you recognize the solution of  If you said  you’re right! We say  is a solution to the equation  because when we substitute  for  the resulting statement is true.
Since  is a true statement, we know that  is indeed a solution to the equation.
The symbol  asks whether the left side of the equation is equal to the right side. Once we know, we can change to an equal sign  or not-equal sign 

### Model the Subtraction Property of Equality

We will use a model to help you understand how the process of solving an equation is like solving a puzzle. An envelope represents the variable – since its contents are unknown – and each counter represents one.
Suppose a desk has an imaginary line dividing it in half. We place three counters and an envelope on the left side of desk, and eight counters on the right side of the desk as in . Both sides of the desk have the same number of counters, but some counters are hidden in the envelope. Can you tell how many counters are in the envelope?
What steps are you taking in your mind to figure out how many counters are in the envelope? Perhaps you are thinking “I need to remove the  counters from the left side to get the envelope by itself. Those  counters on the left match with  on the right, so I can take them away from both sides. That leaves five counters on the right, so there must be  counters in the envelope.”  shows this process.
What algebraic equation is modeled by this situation? Each side of the desk represents an expression and the center line takes the place of the equal sign. We will call the contents of the envelope  so the number of counters on the left side of the desk is  On the right side of the desk are  counters. We are told that  is equal to  so our equation is
Let’s write algebraically the steps we took to discover how many counters were in the envelope.
Now let’s check our solution. We substitute  for  in the original equation and see if we get a true statement.
Our solution is correct. Five counters in the envelope plus three more equals eight.

### Solve Equations Using the Subtraction Property of Equality

Our puzzle has given us an idea of what we need to do to solve an equation. The goal is to isolate the variable by itself on one side of the equations. In the previous examples, we used the Subtraction Property of Equality, which states that when we subtract the same quantity from both sides of an equation, we still have equality.
Think about twin brothers Andy and Bobby. They are  years old. How old was Andy  years ago? He was  years less than  so his age was  or  What about Bobby’s age  years ago? Of course, he was  also. Their ages are equal now, and subtracting the same quantity from both of them resulted in equal ages  years ago.

### Solve Equations Using the Addition Property of Equality

In all the equations we have solved so far, a number was added to the variable on one side of the equation. We used subtraction to “undo” the addition in order to isolate the variable. 
But suppose we have an equation with a number subtracted from the variable, such as  We want to isolate the variable, so to “undo” the subtraction we will add the number to both sides. 
We use the Addition Property of Equality, which says we can add the same number to both sides of the equation without changing the equality. Notice how it mirrors the Subtraction Property of Equality.
Remember the  twins, Andy and Bobby? In ten years, Andy’s age will still equal Bobby’s age. They will both be 
We can add the same number to both sides and still keep the equality.

### Translate Word Phrases to Algebraic Equations

Remember, an equation has an equal sign between two algebraic expressions. So if we have a sentence that tells us that two phrases are equal, we can translate it into an equation. We look for clue words that mean equals. Some words that translate to the equal sign are:
1. is equal to
2. is the same as
3. is
4. gives
5. was
6. will be
It may be helpful to put a box around the equals word(s) in the sentence to help you focus separately on each phrase. Then translate each phrase into an expression, and write them on each side of the equal sign.
We will practice translating word sentences into algebraic equations. Some of the sentences will be basic number facts with no variables to solve for. Some sentences will translate into equations with variables. The focus right now is just to translate the words into algebra.

### Translate to an Equation and Solve

Now let’s practice translating sentences into algebraic equations and then solving them. We will solve the equations by using the Subtraction and Addition Properties of Equality.

### Key Concepts

1. Determine whether a number is a solution to an equation.

           If it is not true, the number is not a solution.

2. Subtraction Property of Equality


3. Solve an equation using the Subtraction Property of Equality.

4. 
Addition Property of Equality
 
5. Solve an equation using the Addition Property of Equality.


### Practice Makes Perfect

Determine Whether a Number is a Solution of an Equation 
In the following exercises, determine whether each given value is a solution to the equation.
Model the Subtraction Property of Equality 
In the following exercises, write the equation modeled by the envelopes and counters and then solve using the subtraction property of equality.
Solve Equations using the Subtraction Property of Equality 
In the following exercises, solve each equation using the subtraction property of equality.
Solve Equations using the Addition Property of Equality 
In the following exercises, solve each equation using the addition property of equality.
Translate Word Phrase to Algebraic Equations 
In the following exercises, translate the given sentence into an algebraic equation.
Translate to an Equation and Solve 
In the following exercises, translate the given sentence into an algebraic equation and then solve it.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?
