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Algebraic Equations and Inequalities

# GED Mathematical Reasoning: Equations I

• An equation contains an equal sign.
• To SOLVE an equation means to find the value of the variable that makes the equation TRUE. This value is called the SOLUTION.
• To solve an equation, use the concept of OPPOSITE OPERATIONS to get the variable by itself on one side of the equal sign. Remember: what you do to one side of the equation, you must also to do to the other side to keep the equation BALANCED.
• Always check your solution to make sure the result is a true statement.

Example 1

Translate the following statement into a mathematical equation: a number plus sixteen equals twenty-one

An equation states that two things are equal. You can identify an equation very easily because an equation contains an equal sign. An algebraic equation may contain numbers, variables, operations, and it will – of course – contain an equal sign. As you know, a variable is a letter that represents a number that we don’t know. The answer to an equation is called the solution. The solution is the number that the variable needs to be in order to make the statement TRUE.

Going back to Example 1, when translating words into mathematical statements you are already familiar with words like “sum,” “minus,” “of,” and “quotient” which imply addition, subtraction, multiplication, and division respectively.

minus subtraction
[of multiplication
quotient division

And there are words that mean equality, too. Words like:  equals, is, total, result, same as, and equivalent all imply the equal sign.

” = “: equals, is, total, result, same as, equivalent

So to translate the words in Example 1 into a mathematical equation, we’ll take it one word or phrase at a time.

The phrase “a number” implies a number we don’t know so we’ll represent it using a variable. You may use any letter you wish, but it is common to use a letter like “x” or “n.”

The word “sixteen” is written as the number 16.

The word “equals” translates into the equal sign.

And the word “twenty-one” is written as the number 21.

So the equation is:

Example 2

Given the equation: , is  a solution?

In this example, we’re being asked to determine if  is a solution to the equation

In other words, if we allow “x” to be “2” – is the statement TRUE?

When we replace the variable “x” with the number 2, we get: two plus four equals nine

Two plus four equals 6, which is NOT equal to 9. Therefore is NOT a solution because when we substitute x for 2, the result is NOT a true statement.

Moving on from here, a logical question to be thinking is: So what IS the solution to the equation in Example 2? Ask yourself:  What number added to 4 equals 9? The answer, or solution, is 5 since 5 plus 4 equals 9 and “9 equals 9” is a TRUE statement.

Example 3

Solve the equation:

First, focus on the side of the equal sign containing the variable. In this case, the variable “n” is on the left side of the equal sign.

Our goal is to get that variable by itself by getting rid of the subtraction 5, so to speak, and we will do so using opposite operations. The operation shown on the left side between the “n” and the 5 is subtraction. Therefore, we will use the operation that is opposite of subtraction – which is addition.

There is one more thing I’d like to mention before we go any further. When you perform an operation to one side of the equation, you must do the SAME thing to BOTH sides of the equal sign.

Think of it this way: an equation is like a scale, where the equal sign is the pivot point. When working with equations, it is important to ALWAYS keep the scale balanced – if you add to one side, you must add the same number to the other side; if you subtract from one side, you must subtract the same number from the other side, too.

Some students find it very helpful to draw a line down their paper from the equal sign to keep each side of the “scale” or equation separate. I encourage you to do the same.

Back to the example.

When we add 5 to both sides, on the left side we have “n” minus 5 plus 5.  Minus 5 plus 5 undoes each other, leaving just “n.”

On the right side, 12 plus 5 equals 17. So we have, for the solution:

As a final thought, it is a good habit to always check your solution. To do so, go back to the original equation and substitute the variable with the number solution and verify that it results in a TRUE statement.

So we’ll go back to our equation of: and replace “n” with 17.

Does 17 minus 5 equal 12? The answer is YES! “17 minus 5 equals 12” results in the statement “12 equals 12,” which IS a TRUE statement.

Example 4

Solve the equation:

Notice that the variable “x” is on the left side.

Our goal is to get the variable “x” by itself by getting rid of the seven, so to speak.

The operation shown on the left side between the “x” and the 7 is multiplication. Therefore, we will use the operation that is opposite of multiplication – which is division. We will divide BOTH sides by 7.

One the left side, the multiplication and division undo each other, leaving just “x.”

On the right side, 42 divided by 7 equals 6.

To check, we’ll go back to our original equation and replace “x” with 6. Since 7 times 6 does, in fact, equal 42 and a true statement results, our solution checks out!

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