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Balance as a model of an equation

This article discusses how to use the balance as a model or illustration of simple equations for students who are just starting to study equations.

An equation is basically saying that two things are EQUAL.  Since in a balanced situation the two sides of the balance hold equal weight, we can model simple equations with a balance.

In the pictures balls represent ones, and the block represents the unknown x.  To find out what the block weighs, you can

  • add the same amount (balls or blocks) to BOTH sides
  • take away the same amount from BOTH sides

That way you will maintain the balance (or the equality in the equation). 

 

If this is a balanced situation...

x + 3 = 5
 

        

...so is this!  
(We took away three balls from BOTH sides.)

x = 2

 

Take away two blocks (or two x's) from both sides. The balance will stay balanced.

3x + 2 = 2x + 6
 

Take away 2 balls from both sides. The balance will stay balanced.

x + 2 = 6
 

     

Here is the solution!

x = 4

 
Without the balance model, the solving process looks like this:

3x + 2
-2x      

=
 
 2x + 6
-2x
        
(take away 2x from both sides)

x + 2
-2

 =
 
 6
-2
 
(take away 2 from both sides)

 x

 = 4

Dividing

In some situations you have to divide both sides of the equation by the same number.  When is that?  It's basically when you have the happy situation where there are ONLY x's (blocks) on one side but there's more than one.

 

 

If you take half of of the things on the left side away, and similarly take half of the things on the right side, the balance will stay balanced.

2x = 8
 

     

x = 4

 

Think about it! 

If this is a balanced situation...

3x = 9
 

     

...so is this! (and vice versa)
We simply divided both sides
by 3.

x = 3

 

Combining the operations

The legal operations thus are:

  • add the same amount to both sides (either x's or ones)
  • subtract the same amount from both sides (either x's or ones)
  • divide both sides by the same number (but not by zero)
  • multiply both sides by the same number (but not by zero)

(There are others too but those are not needed in these simple equations.) 

The goal  is to FIRST add and subtract until we arrive to a situation where on one side there are ONLY x's (blocks) and on the other side there are ONLY ones (balls).

Then when you have only blocks on one side, if you have more than one, you need to divide so as to arrive to the situation with only one block on the one side, which is the solved equation!

Multiplying both sides can occur if you have a fractional block (less than one block) on one side.  For example, the equation  1/4x = 13  is solved by multiplying both sides by 4.  You can for example let your students draw the situation   1/2x + 14 = 20   and they can still solve it with the balance.  And more advanced students can ponder what to do about the equation  2/3x = 12.


Example of both subtracting and dividing

In this example we use all the abovementioned operations: taking away from both sides of the equation and dividing the equation by the same number.

4x + 2 = 2x + 5

First we get rid of the blocks on the right side
 - take away two blocks from both sides.

2x + 2 = 5

Then we get rid of the balls on the left side and
so we take away 2 balls from both sides.

2x = 3

Now there are ONLY blocks on one side and
ONLY balls on the other.  To know what 1 block
weighs, we take half of both sides.

x = 1 1/2

The solution is that 1 block weighs 1 1/2 balls.

Try substituting this value x = 1 1/2 into the original equation 4x + 2 = 2x + 5 and check if the equation becomes true!

These equations are simple enough that you can solve them using this balance model.   ALWAYS check your solution by substituting to the original equation.

  1. 2x + 3 = 5
  2. 2x + 5 = x + 9
  3. 3x + 2 = 2x + 4
  4. 3x + 3 = 5 + x
  5. 5x + 4 = 3x + 6
  6. 6x + 2 = 3x + 6
  7. 6x + 3 = 2x +

Continue to Negative terms in an equation

How to teach equation solving: example (from my blog)



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