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Two Step Equations

Introduction

The saying “x marks the spot” has its origins in the early days of newspapers where the location of a crime would be marked with an “x” on a map.  In movies, the letter “x” is often the location of a hidden chest of gold on a treasure map.  To find the treasure, a pirate or treasure hunter must carefully follow the steps in precise order to end up at the correct location.

Just as the “x” on a treasure map can be found by following the directions in order, an equation can be solved by following the correct mathematical steps in order.

 

To understand how equations are solved, it helps to understand how they are constructed.  We can construct a two step equation as follows.

          First, pick a number.  For this example, we can choose the number 7.

              Step 1: Multiply our number by 5.  The result is 7 ´ 5 = 35.

              Step 2: Subtract 11 from the result of step 1.  Our final result is 35 – 11 = 24.

 

Now suppose we tell a friend the steps we followed to get our answer of 24.  Multiplying our original number by 5, then subtracting 11 can be represented by the following equation:

5n – 11 = 24.

The equation can be solved using the two steps to the right.          Two_Step_Equations_visuals_1


Lesson

In the previous lesson, you learned how to solve one step equations.  Before reading about the best way to solve two step equations, take a look at the similarities and differences of the following two solutions.

                One Step Equation                                      Two Step Equation

Two_Step_Equations_visuals_2                     Two_Step_Equations_visuals_3                                

 

Each of the equations above is solved by performing the same operation to each side.  In the one step equation, this process is repeated once while in the two step equation it is repeated twice.

 

In the two step equation above (at right), notice that the two steps which must be performed are subtracting 5 and dividing by 8.  The order does matter here... you must subtract five first and then divide by 8 second. Performing the steps in the correct order is important because if done backwards your answer will be wrong (except for some cases where you'd get the same answer either way).  This concept is important, so here are two explanations for you to consider:

 

Explanation #1  (Mathematical)

Consider the left side of the two step equation: 8b + 5.  According to the order of operations, “8 times a number” must be done before the “number plus five” because multiplication is always done before addition.  Since the problem is originally constructed with multiplication before addition, solving (or deconstructing) the problem is done by doing the opposite operations and in reverse order.

Problems that require two steps to be done are most easily done by doing the steps in the following order:

      1st – Undo the addition or subtraction

      2nd – Undo the multiplication or division

 

Two_Step_Equations_visuals_14The diagram to the right shows a person moonwalking (walking backwards) across the order of operations.  The only operations you will see in this lesson are multiplying, dividing, adding, and subtracting.


 

Explanation #2 (Common sense)

Consider the two-step task of taking off a coat and putting it in the closet.  This task requires two steps.

      Step 1:  Take off the coat

      Step 2:  Put it in the closet

 

Now suppose that you want to reverse the task.  The coat is now in the closet, so you must first take the coat out of the closet, then put it on.  Compare these two steps to the steps above:

      New step 1:  Take coat out of the closet  (reverse of step 2 above)

      New step 2:  Put the coat on  (reverse of step 1 above)

 

Solving a two-step equation has a lot of similarities with putting the coat back on.  The second step must be reversed, then the first step must reversed to get back to the original situation.

 

Here are some more examples of two step equations.


 

Example 1:  3f + 5 = 20                                     

 

Solution:

 Two_Step_Equations_visuals_5

 

 

Example 2:  5 – 2g = -13

 

Solution:

Two_Step_Equations_visuals_6

Review

 

1)  8 + 5c = 33                                  

2)  -18x + 30 = -150 

3)  5x + 21 = 61

4)  3(x + 2) = -27

 

 

Scroll down for answers:

 

 

 

 

 

 

 

Answers:

 

1) Two_Step_Equations_visuals_7

 

 

 

2) Two_Step_Equations_visuals_8

 

 

 

3) Two_Step_Equations_visuals_9

 

 

 

4) Two_Step_Equations_visuals_10

 


 

Related Links:

Looking for a different lesson on solving equations?  Try the links below.

 

Related Lessons

 

Looking for something else?  Explore our menu of general math or algebra lessons.

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