Solving One-Step Equations with Addition and Subtraction | Algebra How To Help

Solving One-Step Equations

It does not matter which textbook your school has chosen to use, nor does it really matter the age of the text. Solving multi-step equations will be apart of what you have learned. I work with Prentice Hall's Algebra 1 (2009) and in the years before that I worked with Glencoe's Algebra 1 (1999). I tell you what books I have used because that should give you a clear picture that solving equations is an important part algebra and learning how to solve one-step equations is the building block to solving two-step and multi-step equations. Remember, the goal of solving multi-step equations is to isolate the variable correctly to get the answer.

Isolating the Variable

Example 1

To solve equations means to "isolate the variable", which means to get the 'letter on one side of the equal sign'. As in example 1 to the right, equation (1) is x + 4 = -6. On the left of the equal sign is x + 4 and on the right is -6. To 'get the letter by itself', undo the add 4 by subtracting 4 from both sides as shown in red. Equation (2) gives the variable now isolated with a value equal to -6. From this example, I hope you understand how I will tie the examples into what I am blogging about. Please notice how I use numbers next to the equations.


Solving One Step Equations with Addition and Subtraction

Example 2

Example 1 is a perfect example of solving a equation with subtraction. The reason subtraction was used is because the original equation involves addition. The focus is usually what operation, addition or subtraction, is between the variable and the number. This strategy will work most of the time, but not always.

Example 2 is another equation in which the focus can be on the operation between the variable and the number. The focus can be on the the sign between the number and variable when the number is positive. As in example 2, since 6 is added to the variable, to get rid of the 6, it is subtracted from the 6 on the left and it is subtracted from the right to maintain the balance of the equation. Because the 6 and -6 on the left are opposites, those numbers combine to give 0 and the variable is isolated. Another way teachers will describe what happens on the left is to say "the 6's cancel".

Example 3

Example 3's original equation start with subtracting a positive 6 from the variable. As in example 2, in equation 1, the number on the same side as the variable is positive, so the focus can be on the operation separating the variable and the number, in this case it is subtraction. In the pink, adding 6 to both sides is shown. The -6 and plus 6 cancel on the left and on the right, -11 + 6 is equal to -5 and I conclude equation 2 as x = -5.Just as I did in both examples 1 and 2, I performed the operation to both sides of the equation. In this example, I added 6 both sides by using the addition property of equality. As you continue with this lesson, the original equation will need to be modified before the focus of the opposite operation that is between the variable and the number can be addressed.

Addition is Subtraction and Subtraction is Addition

What? This topic can be be very confusing if you are unaware of two facts:

1. Adding a negative number is the same as subtracting a positive number
2. Subtracting a negative number is the same as adding positive number

Example 4 Incorrect and Correct Way To Solve an Equation

In the above image, I model how to solve the same equation two different ways. I solve equation 1by subtracting 4 from both sides. This is in correct because of the double sign between the variable and the number. The double sign means I should employ one of the rules listed above. In this case a negative 4 is being added, by rule 1, I should think of that as subtracting a positive 4. You can see how I changed my work with equation 1 as I changed it to equation 2. Once the equation is written with only one sign the focus can be on using the opposite operation, the opposite of subtracting 4 is adding 4 and that work is done in red.

Example 5

In example 5 the original equation is x + (-2) = 8. From equation 1 to equation 2 I apply rule #1 from above. Now that equation is written with only one sign between the variable and the number, using the opposite operation now applies, since 2 is being subtracted in the equation, I must add 2 to both sides of the equation. On the left, the -2 and +2 cancel because they are opposites. On the right, 8 + 2 is 10, therefore x = 10.

If it is obvious to you that you need to add to solve this equation, then rewriting the equation as equation 2 is not necessary to find the correct solution.

Example 6

The final example, rule #2 is illustrated. Equation 1 is x - (-5) = 7. Notice the "double negative" between the variable and the number. Some teachers, including will say, "a double negative makes a positive." We say that because it does and it offers an easy way to remember rule number 2. After using rule #2, the equation can be solve by subtracting 5 from both sides. This isolates the variable and gives x = 2.

Using the below video will give you some interactive practice with solving one-step equations with addition and subtraction.