Introductory Algebra

1.3 – The Multiplication Principle of Equations

The Multiplication Principle of Equations

Imagine again our fulcrum and lever from the previous section:

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Instead of adding a third shape, lets add another square to the left side of the fulcrum. What happens?

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Similarly to adding the circle, it will fall to the left. But if we add a triangle to the right, since the triangle is equal to the square, we reach equilibrium again:

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In this situation, we have multiplied each side by two. This brings about a similar, equally important property of equations:

The Multiplication Principle of Equations
The Multiplication Principle of Equations states that if you multiply both sides of an equation by the same value, you maintain the equation. Mathematically, this is:
If a=b

Then ac=bc

This property is particularly useful in equations such as 3x=9. We saw that in the additive property, we could also subtract numbers from each side. In the multiplication property, we may also divide numbers from each side. For this section, we will primarily focus on the fact that we can divide each side by the same number. We will revisit this property again in a later section for multiplication specific application.

 

Solve the equation 3x=9
Let’s divide each side by 3:
3x=9

\frac{3x}{\color{red}3}=\frac{9}{\color{red}3}

x=3

 

Solve 4x=16
4x=16

\frac{4x}{\color{red}4}=\frac{16}{\color{red}4}

x=4

In many applications, we will need to use the Addition and Multiplication principles together:

Solve: 4x-2=6
First use the addition property to isolate the variable term:
4x-2=6
\underline{\phantom{4x}+2 +2}

4x=8

Then use the multiplication principle:
\frac{4x}{\color{red}4}=\frac{8}{\color{red}4}

x=2

And we might also need to distribute and simplify as well:

Solve: 3(x+2)=9
3(x+2)=9

3x+6=9
\underline{\ \ \ \ -6 -6}

3x=3

\frac{3x}{\color{red}3}=\frac{3}{\color{red}3}

x=1

 

Solve: 3(x+2)+5=2(3x-5)

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\frac{21}{\color{red}3}=\frac{3x}{\color{red}3}

7=x

 

Solve: 2(x+3)+3(x-2)=2(x-4)+x-8
2(x+3)+3(x-2)=2(x-4)+x-8

2x+6+3x-6=2x-8+x-8
5x=3x-16
 
2x=-16
 

x=-8

 

Solve: \frac{2}{5}x=\frac{7}{10}
Here, we can divide each side by the fraction \frac{2}{5}, but recall that dividing by a fraction is the same as multiplying by its reciprocal. Thus, we can multiply both sides by the fraction \frac{5}{2}:
{\color{red}\frac{5}{2}\cdot}\frac{2}{5}x=\frac{7}{10}{\color{red}\cdot\frac{5}{2}}

x=\frac{7}{4}