## Solving Algebraic Equations – Think Opposite!

When solving basic algebraic equations, always think “Do the opposite.”

This means that you will be doing the opposite or inverse operation in order to move numbers across the equals sign.

Let’s have a look at a few examples. I have tried to show each step separately. As you become more proficient, several steps can be combined, but I wanted you to have as clear a picture as possible.

To solve an algebraic equation, you need to find the value of the variable. The variable is represented by a letter – often “x” but any letter will do.

For example:  x + 9 = 14   or   s – 18 = -12

The first thing you should think is, “I have to get the variable (letter) by itself. Although it doesn’t matter, by convention try to get the variable on the left side of the equal sign and the numbers on the right side.

This is where “do the opposite” applies.

Let’s look at a simple example.

A.      82 + x = 246

In order to get x by itself, I need to move the 82 from the left side to the right side. To do this, I am going to complete the opposite operation – subtraction. If I subtract 82 from 82, I will be left with just the variable (x); however, because this is an equation, I must do the same operation on both sides to keep it equal. (Think of one of those old balance scales. If I add 2kg on one side and not the other, it will no longer be equal. It will not be an equation.) Now we have the following:

82 – 82 + x = 246 – 82

x = 164

That’s it!

Let’s look at a few more examples.

B.       x + 146 = 669

x + 146 – 146 = 669 – 146

x = 523

Here are a couple of subtraction examples.

C.      x – 344 = 581  (Now you must add to eliminate -344 on the left.)

x – 344 + 344 = 581 + 344

x = 925

Just as addition and subtraction are inverse or opposite operations, so are multiplication and division. Let’s have a look at a few examples.

D.      6 x s = 24 or 6s = 24       (6s is the same as 6 x s.)
6s = 24        (We need to divide both sides by 6 because 6 ÷ 6 = 1 which will leave us with 1s usually simply written as s.)
6s ÷ 6 = 24 ÷ 6
s = 4

Here is a division equation.

E.       s ÷ 12 = 2    (We need to do the opposite – multiply.)

s x 12 = 2 x 12

s = 24

Of course all of the examples above are quite simple and get easily be solved without going through these steps. The idea is to get used to following the procedure so that when faced with more challenging questions, you will be able to easily follow the steps to the solution.

Here are a couple more difficult questions dealing with larger numbers and both positive and negative signs.

F.       m + m – 396 = 463 + 55           (Do the math on each side first.)

2m – 396 = 518                         (Now begin doing the opposite.)

2m -396 + 395 = 518 + 396

2m = 914                                  (Remember 2m is the same as 2 x m.)

2m ÷ 2 = 914 ÷ 2

m = 457

Let’s try another one.

G.      -62 + 13 + m = -3m – 45

-49 + m = -3m – 45

-49 + 49 + m = -3m – 45 + 49

m = -3m + 4

m + 3m = -3m + 3m + 4

4m = 4

4m ÷ 4 = 4 ÷ 4

m = 1

Another multiplication example.

H.      5/7r – 3 = 2                    (Move the non-variable number first.)

5/7r – 3 + 3 = 2 + 3

5/7r = 5                          (5/7r is the same as 5/7 x r – so we must divide.)

5/7r ÷ 5/7 = 5 ÷ 5/7     (Remember to multiply by the reciprocal.)

r = 7

The final example for now.

I.        m/-10 = 12           (Currently it is division on the left – so we multiply.

m/-10 x -10 = 12 x -10  (m/-10 x -10/1 = -10/-10 which is 1 = 1m or m.)

m = -120

If you want to be sure that you have the correct answer, “plug” your answer into the original equation.

For example, working on the last two questions.

H.      5/7 (7) – 3 = 2

35/7 – 3 = 2

5 – 3= 2

2 = 2                      check!

I.        -120/-10 = 12

12 = 12                  check!

I hope this has helped clarify a few things in terms of solving algebraic equations.

Of course, I recognize that when we learn or relearn something, many more questions can arise. For example, fractions, order of operation, division, and signed numbers rules can often cause grief.

If you have any questions, please let me know. Also, don’t forget that personalized programs are only an e-mail away!

This week’s video – Solving Algebraic Equations

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