The Addition Property of Equality allows us to solve addition & subtraction equations. This property also specifies that the equation should have the same solution after each operation you perform. $ \begin{align*} x - 1 = 5 \\ x - 1 + 1 = 5 + 1 \\ x = 6 \\ \\ x - 1 = 5 \\ (6) - 1 = 5 \\ 5 = 5 \end{align*} $ In the above example you can see we solved an equation by using the Addition Property of Equality and checked our solution afterwards. The **Addition Property of Equality** tells us that we can add or subtract any value to or from both sides of an equation without changing the solution. Additionally, we need to think about two additional properties that we learned in pre-algebra. First, recall the additive inverse property. This property states that any number plus its opposite results in 0: $5 + (-5) = 0$ $17 + (-17) = 0$ Second, recall that zero is the additive identity. This means that any number plus $0$ leaves the number unchanged: $131 + 0 = 131$ $-25 + 0 = -25$ --- #### Simple Equation Walkthrough: $x - 2 = 7$ Its pretty obvious that $x=9$, this approach works well for simple equations, but as they get more nested and complex, guessing and testing many potential solutions can become extremely dubious. Our goal as shown in previous notes, is to isolate variables, by isolating a variable to one side, we exclaim that the other side must be the value for that variable. The way we solve this equation will be by using the **additive inverse property**. In our case for $x - 2 = 7$ we'll add 2 to both sides: $x - 2 = 7$ $x - 2 + 2 = 7 + 2$ $x = 9$ Here's a more complex example: $5 + x - 3 = 7 + 2x$ $2 + x = 7 + 2x$ $2 + x - 2 = 7 + 2x - 2$ $x = 5 + 2x$ We're using the [[Multiplication Property of Equality]] here to switch signs, for now just take it as a given, but the note above explains it in full detail. $-x = 5$ $x = -5$ $5 + x - 3 = 7 + 2x$ $5 -5 - 3 = 7 + 2(-5)$ $-3 = -3$ --- #### Proof for Addition Property of Equality Given: - $a = b$ Now, let’s add the same value $c$ to both sides: - $a + c = b + c$ Since $a = b$ (given) we can rewrite the equation as such: - $b + c = b + c$ The equation is identical on both sides indicating that we've maintained the solution. ✅ If $a = b$ **then** $a + c = b + c$ for any real number $c$ --- # Practice Tests - [[Addition+Property+of+Equality+Bonus+Practice+Test.pdf]] - [[Addition+Property+of+Equality+Practice+Test.pdf]]