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2.3.3 The Multiplication Property of Equality

The Addition Property of Equality says we can add the same number to both sides an equation and it will stay equal.  There's a law just like this for multiplication.  You can also multiply both sides of an equation by the same number and the equation will stay equal.

Equations are kind of like the bars used in weight lifting.  They only work if you have the same amount on each side.

For the weights on the bar, we could say:

l = the amount of weight on the left side

r = the amount of weight on the right side 

For the weight set up to work right, the weight on the left side has to equal the weight on the right side.  So

If we triple the amount of weight on each side, the set up will still work (as long as we can still lift it).  In math terms, we multiply both sides of the equation by 3.  So 

This rule also works for division (as long as we don’t try to divide by 0).  We can divide the amount of weight on each side of the bar by the same number and it will still work.

The problem below shows another example how the Multiplication Property of Equality can be used. 

Example 2.10:  In a baseball game, Scott batted .500 with 6 at-bats (excluding walks).  How many hits did he get during the game?

Solution:  In this problem, we need to find the number of hits.  We'll say: 

h = the number of hits during the game

The formula for batting averages is:

where    a = the number of at-bats during the game (excluding walks) = 6

            b = the batting average during the game = .500 = 1/2

From the equation above:

Like the examples before, the key to solving this problem is to get the variable we want (in this problem, h) by itself on one side of the equation.  To do this we need to get rid of the 1/6 on the right side of the equation.  This is done by multiplying both sides of the equation by the Multiplicative Inverse or reciprocal of the number you’re trying to get rid of.  The reciprocal of a number is the number to multiply it to 1.  The reciprocal of 1/6 is 6.  The Multiplication Property of Equality allows us to multiply both sides of the equation by 6. 

Therefore,   h = 3.

Scott gets 3 hits during the game.

As we saw before, the Multiplication Property of Equality also works with division.  This is because dividing both sides of an equation by a real number is the same as multiplying the equation by the number's reciprocal.

This property shows why the algebraic laws are limited to real numbers.  If we were to divide both sides of an equation by 0, it would be the same thing as multiplying it by 1/0, which is an unreal number.  Division by zero is not permitted in math.

Note - Again we used a very simple but important law of algebra when we said 1 x h = h.  This property is known as the Multiplicative Identity Property.

The Multiplicative Identity Property For any real number a a x 1 = a