Balancing Method

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 picture below, each circle represents, “one” (1), and the block represents the unknown “x“. To find out what the block weighs, you can do the following:

That way both sides will maintain the balance or “equality”

Example 1:

The diagram shows a balanced lever. The objects on both sides of the fulcrum (triangle) are equal.

Represent the unknown with the letter x

A block \( \scriptsize \boxed {?} \) Represents x which is unknown

A circle \( \Large\circ \) Represents 1 

∴  \( \scriptsize x \: + \: 3 = 5 \)

The lever will remain balanced if we removed 3 circles from each side.

We took away three circles from BOTH sides to make the lever balanced.

This reads as one block = two circles.

i.e.  x  = 2

If x = 2 this can also mean that a block is equal to two circles, which means they both have the same weight.

Therefore the diagram can also be drawn like this;

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

⇒ \( \scriptsize x \: + \: 3 = 5 \)

⇒ \( \scriptsize x \: + \: 3 \: – \: 3 = 5 \: – \: 3 \)

⇒ \( \scriptsize x = 2 \)

Example 2

⇒ \( \scriptsize 3x \: + \: 2 = 2x \: + \: 6 \)

First, take away two blocks (Two x’s) from both sides.

Then we have \( \scriptsize x \: + \: 2 = 6 \)

Take away two circles from both sides

The balance will stay balanced.

i.e. x  =   4

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

\( \scriptsize 3x \: + \: 2 = 2x \: + \: 6 \)

(Take away 2x from both sides)

\( \scriptsize 3x \: + \: 2 \; – \: 2x = 2x \: + \: 6 \; – \: 2x \)

Collect like terms

\( \scriptsize 3x \; – \: 2x \: + \: 2 = 2x \; – \: 2x \: + \: 6\)

\( \scriptsize x \: + \: 2 = 0 \: + \: 6\)

\( \scriptsize x \: + \: 2 = 6\)

Take away 2 from both sides

\( \scriptsize x \: + \: 2 \; – \: 2 = 6 \; – \: 2\)

\( \scriptsize x \: + \: 0 = 4\)

\( \scriptsize x = 4\)