The Multiplication Property of Equality allows us to solve multiplication & division equations. This property also specifies that the equation should have the same solution after each operation you perform. $ \begin{align*} 2x &= 10 \\ \frac{2x}{2} &= \frac{10}{2} \\ x &= 5 \\ \\ 2x &= 10 \\ 2(5) &= 10 \\ 10 &= 10 \\ \\ \frac{x}{9} &= 17 \\ \frac{x \cdot 9}{9} &= 17 \cdot 9 \\ x &= 153 \\ \\ \frac{153}{9} &= 17 \\ 17 &= 17 \end{align*} $ In the above examples we solved both equations by using the Multiplication Property of Equality. We also proved these solutions are valid by checking them right after solving them. Before we dive into the **Multiplication Property of Equality** we'll quickly go over 2 pre-algebra properties. --- #### Multiplicative Inverse Property: This property tells us that a non-zero number multiplied by its inverse (recipricol) always returns 1: $ \frac{3}{5} \cdot \frac{5}{3} = 1 $ $ 7 \cdot \frac{1}{7} = 1 $ --- #### Multiplicative Identity Property: This property tells us that multiplying a number by one leaves the number completely unchanged. $6 \cdot 1 = 6$ $-\frac{2}{9} \cdot 1 = -\frac{2}{9}$ The **Multiplication Property of Equality** tells us that we can multiply or divide both sides of an equation by the same non-zero number and not change the solution. This is similar to the **Addition Property of Equality**. This also supposes you understand that multiplication ($\times$) and division ($\div$) are inverse operations, similar to how addition ($+$) & subtraction ($-$) are. $3 \cdot 7 = 21$ $21 \div 3 = 7$ $21 \div 7 = 3$ Here is an example with an explanation: $ \begin{align*} 2x &= 10 \\ \frac{2x}{2} &= \frac{10}{2} \\ x &= 5 \\ \\ 2x &= 10 \\ 2(5) &= 10 \\ 10 &= 10 \end{align*} $ In this example x is multiplying by 2. We can do the inverse based on the multiplication property without changing the solution. This example states the obvious, as $2 \times 5 = 10$. We also checked the result $-$ resulting correct! --- #### Nuances: - You can't divide or multiply both sides of an equation by zero as it assigns both sides to undefined / 0 --- #### Proof for the Multiplication Property of Equality We begin with the assumption: - $a = b$ Let $c \in \mathbb{R}$ and $c \ne 0$ Now, multiply both sides by $c$: - $ac = bc$ Since $a = b$ (given) we can rewrite the equation as such: - $bc = bc$ ✅ If $a = b$ **then** $ac = bc$ for any real (non-zero) number $c$. This is valid because multiplying both sides of an equation by the same non-zero number preserves equality. Now apply the **Multiplicative Inverse Property** by multiplying both sides by $\frac{1}{c}$: $ \frac{1}{c} \cdot ac = \frac{1}{c} \cdot bc $ Using associativity: $ (a \cdot c) \cdot \frac{1}{c} = (b \cdot c) \cdot \frac{1}{c} $ Apply the **Multiplicative Inverse**: $ a \cdot 1 = b \cdot 1 $ Apply the **Multiplicative Identity**: $ a = b $ ---- # Practice Tests - [[Multiplication+Property+of+Equality+Bonus+Practice+Test.pdf]] - [[Multiplication+Property+of+Equality+Practice+Test.pdf]]