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(Video) The Multiplication Property

by mathman1024

This video by mathman1024 explainins the multiplication property.

$\large{\text{Summary}}$

The Multiplication Property is similar to the addition property where: $$a=b\\ \text{is equivalent to}\\ a\,{\color{#8925AE}{\times\, c}}=b\,{\color{#8925AE}{\times \,c}}\\ \text{or}\\ \frac{a}{\color{#8925AE}{c}}=\frac{b}{\color{#8925AE}{c}}$$ Just like the property of addition, whatever you do to one side you have to do to the other.

He shows this in the example: $2x=10$. $$\begin{align} 2x&=10\\ \frac{2x}{\color{#8925AE}{2}}&=\frac{10}{\color{#8925AE}{2}}\\ x&=5 \end{align}$$ Because you want to get $x$ by itself you divide its side by $2$. But then you also have to divide the other side by $2$ in order to keep them equal.

Try the last one by yourself and see if you get the same answer.

1. $8x=-60$

Bonus: check your work by plugging in the value for $x$ you solved for back into the equation!

The principle of equality means that the information on either side of an equals sign must be worth the same amount. Think of it in terms of weight: if we have five apples on one side of a scale, and two melons on the other side, the scale will be balanced:

We can write this as: $5A = 2M$,

where the nonzero variable $A$ represents an apple and $M$ represents a melon.

If we multiply the number of apples by $3$ and the number of melons by $3$, we won't be adding the same number of fruits to both sides, but relative weight should stay the same—they should both weigh $3$ times as much.

The multiplication and division properties of equality state that if you multiply or divide both sides of an equation by the same number, then both sides are still equal. This is similar to the Addition and Subtraction Properties of Equality.

In other words, if $a+b = c$, then $d(a + b) = d(c)$.

Example: When we solve $x+3=9$, you get $x=6$. Let's see what happens when you multiply both sides by $2$. $$\begin{align} 2(x+3)&=2(9)\\ 2x+6&=18\\ 2x+6-6&=18-6\\ 2x&=12\\ \frac{2x}{2}\ &=\frac{12}{2}\\ x&=6 \end{align}$$

If you're curious, check out some other properties of equality, like the reflexive, symmetric, and transitive properties!

You'll use a lot of these properties when you're solving linear equations, quadratic functions, inequalities, and absolute value expressions.