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Equations w Variable on Both Sides

Introduction

Equations_w_variable_on_both_sides_visuals_1
When solving a one step equation, two step equation, or multi-step equation, the goal is to isolate the variable on one side.  With some practice, this is easily done by manipulating the equation.  One thing that all of these equations have in common is that the variable term(s) are always one side of the equation, leaving the other side as just numbers.  Notice that each of the equations to the right has the variable terms on the left hand side of the equation.

 




Equations_w_variable_on_both_sides_visuals_2Each of the equations above can be solved by manipulating the equation using mathematical properties until the variable equals the solution.  An example of solving a multi-step equation is shown to the left.  Pay special attention to the red variable terms.  Notice that the variable terms are both on the left side of the equation, so they can be combined together into a single variable term by adding them together.


How do you think you would solve an equation with a variable term on each side?  An example of this is below.

Equations_w_variable_on_both_sides_visuals_3

Guess and check would be an acceptable way to solve the equation.  A good way to think of it would be to translate the problem from math into English.

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If a number and five is the same as twice the number, trial and error can be used to figure out that the number must be 5.  Five plus five is ten, which is the same as two times five.

Before moving to the lesson, use trial and error to find the answer to the following problems.

     n + 30 = 4n                          3n = n + 16




Lesson

When solving a problem that has a variable on each side, an easy way to think about solving it is to have the variable stand for a word.  Consider the first equation given in the introduction to this lesson.

Equations_w_variable_on_both_sides_visuals_3

For the variable “n”, simply think of a word that begins with “n” that could be represented by the variable.  Take the word “nickel” for example.  The equation could be rewritten:

a nickel plus five is equal to two nickels.


Equations_w_variable_on_both_sides_visuals_7Think of each side of the problem as an equal group.  The left group is a nickel plus five and the right group is two nickels.  In mathematics, you are allowed to “take away” the same amount from each side of an equation to simplify it, so take away one nickel from each group.  The result is simply that five equals a nickel.


Many learners can solve the above problem in their head.  When solving a more difficult problem, it can be done in much the same way.  Simply look at the variable terms and subtract the smaller one from both sides.

Consider the equation 4f + 16 = 7f – 20.

You may be able to solve this equation in your head if you think about it long enough.  However, writing out your work on paper is the recommended way to do more difficult problems.  Your teacher will be pleased if you can show all the work and you will be able to explain it to a friend if you can demonstrate all your steps.

Equations_w_variable_on_both_sides_visuals_8

The above problem contained the variable term 4f on the left and 7f on the right.  The easiest way to do the problem is to subtract the smaller of the variable terms from each side.  Subtracting the 4f from the 7f yields a difference of 3f.  If you subtract the smaller variable term from the larger variable term, you will end up with a positive variable term as your result.  It is generally easier to work with positive numbers, and if you remember to always subtract the smaller variable term from both sides you will always get a positive result.

Here is a slightly more complicated equation:

5(z + 4) -13 = 2z + 19.

This equation can be solved in much the same way the one above it (4f + 16 = 7f – 20).  The only difference is some simplifying beforehand.

Equations_w_variable_on_both_sides_visuals_9

Equations_w_variable_on_both_sides_visuals_10There may be times where a variable term is negative.  Remember that subtracting a negative is actually the same as adding.  Here is an example of an equation with a negative variable term on one side.



 

Try It

 

Solve the following equations:

4c + 2(3c + 4) = 6c + 44

 

 

 

 


18w + 13 = -12w + 163



 


 

 

Related Links:

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

 

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

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