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).
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.
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.)
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.
- 2x + 3 = 5
- 2x + 5 = x + 9
- 3x + 2 = 2x + 4
- 3x + 3 = 5 + x
- 5x + 4 = 3x + 6
- 6x + 2 = 3x + 6
- 6x + 3 = 2x + 5
Continue to Negative terms in an equation
How to teach equation solving: example (from my blog)