Solving Two-Step Equations | Algebra How To Help

Solving two-step equations at the algebra 1 level is usually pretty easy for most students. That is because it is generally a review of a pre-algebra lesson. Understanding this lesson requires you to understand solving one step equations. If you are looking for lesson on how to solve one-step equations check these two blog posts:

Solving One-Step Equations with Addition and Subtraction

Solving One-Step Equations with Multiplication and Division

Learning how to solve multi-step equations is a building process and solving two-step equations is really the third step. Once solving one-step equations is mastered, solving two-step equations can be learned and mastered.  Finally, learning how to solve multi-step equations can be introduced.

Solving Two-Step Equations

When solving any equation, you must keep in mind the overall goal: isolate the variable on one side of the equation with a coefficient of 1. If that sounds too mathematical for you, how about this: the goal is to get the variable by itself. To achieve this goal, we use a combination of operations to “undo the order of operations”.  I put, undo the order of operations, in quotes, because that is how to solve two-step equations:

1.              Undo any Addition or Subtraction

2.              Undo any Multiplication or Division

As you should know, the last two steps of the order of operations are:

1.              Complete any multiplication or division from left to right

2.              Complete any addition or subtraction from left to right

I like to think of it as put on a shirt and blazer. When I get dressed, I have to put on the shirt before I put on the blazer or I would look kind of weird. After a long day at school and I get home, I take the blazer off first and then my shirt before sitting down and doing homework with my children. Let me show you building an equation to solve.

Building a Two-Step Equation to Solve

Here I am going to build an equation to be solved. I will use the order of operations to build it. See image 1 for an illustration. I start with the equation

x = 6   (1).

Using the multiplication property of equality, I multiply both sides by 3 to give the equivalent equation

3x = 18   (2).

Next I use the subtraction property of equality to subtract 6 from both sides to give

                                 3x – 6 = 12   (3).

Equation (3) is an example of a two-step equation I will be modeling how to solve in this blog post. In this example, I built an equation to be solved. To solve the equations in this post I will be using the opposite of the order of operations.

Example 1 - Solving Two-Step Equations with Integers

See image 2 for an illustration of this explanation. Starting with the equation

-5x + 42 = -8   (1),

the goal is to isolate the variable. The variable is on the left of the equation, therefore, using the reverse of the order of operations, I will have to undo the addition of 42 and the multiplication of -5. First, using the subtraction property of equality, I subtract 42 from both sides of equation (1) to give

                              - 5x = -50    (2).

Following the reverse of the order of operations I next undo the multiply by -5 by dividing both sides of equation (2) by -5 to give the solution

                                 x = 10    (3).

Hopefully, you found this example easy to understand. Use the video below to practice these skills. In the video, I model how to solve several two-step equations. Then, you are asked to solve some problems on your own.

Example 2 – Solving Two-Step Equations with Decimals

See Image 3 for an illustration for this explanation.

Given the equation:

15.61 = -7.43 + 0.2x    (1).

In equation (1), the variable is on the right hand side of the equation, whereas in Example 1 the variable is on the left hand side. This is a minor difference, but there is a more important difference: the order of the variable term and the constant term. Error Alert: because of this switch of the constant term and variable term, many students start this problem by subtracting 7.43 from both sides. THIS IS WRONG!!! The students that make this error get stuck on the sign that is between the constant and the variable term. It is the sign in front of the constant that determines if you add or subtract to both sides.

To solve this equation correctly, I must add 7.43 to both sides of the equation using the addition property of equality to get

23.04 = 0.2x    (2).

Next I divide each side of the equation by 0.2 using the division property of equality to give

115.2 = x    (3).

Finally, I use the symmetric property of equality to turn the equation around so the variable comes first

X = 115.2    (4).

As you can see from the first two examples, whether you are working with integers or rational numbers, the steps for solving two-step equations remains the same. As you will see, adding fractions to the mix does not change the steps.

Use the video below to practice these skills. In the video, I model how to solve several two-step equations. Then, you are asked to solve some problems on your own.

Example 3 – Solving Two-Step Equations with Fractions

(1).

In my 13 years teaching experience, these types of two-step equations cause even some of the best students to shut down. The steps are no different than I used in Examples 1 and 2, but because there are fractions involved, they shut down and do not even try. With today’s technology, everyone should be able to do calculations involving fractions. Back to the algebra…

The first step to solve this equation is to undo the addition by subtracting from each side to give

  (2).

The next step can be done as I model in Example 5 of my post on solving one-step equations with multiplication and division. To get rid of the fractional coefficient, I can multiply both sides of the equation by the reciprocal or I can use two-steps using a combination of multiplication and division. To stay with the theme of using only two-steps total, I will solve this by multiplying both sides by the reciprocal of  which is . This action isolates the variable giving a solution of

  (3).

Use the video below to practice these skills. In the video, I model how to solve several two-step equations. Then, you are asked to solve some problems on your own.

Conclusion – Solving Two-Step Equations

You will know you are working with an equation that involves two-steps because there will be two operations to “undo”. It does not matter what kind of numbers are involved to solve two-step equations:

1.            Undo any Addition or Subtraction 

2.        Undo any Multiplication or Division