## Understanding the Identity: (a-b)-c = (a-c)-(b-c)

This equation demonstrates an important algebraic identity that helps simplify expressions. Let's break down why it holds true.

### The Basics of Parentheses and Order of Operations

In mathematics, parentheses signify that the operations within them should be performed first. Remember the order of operations (PEMDAS/BODMAS):

**P**arentheses /**B**rackets**E**xponents /**O**rders**M**ultiplication and**D**ivision (from left to right)**A**ddition and**S**ubtraction (from left to right)

### Expanding and Simplifying

Let's expand both sides of the equation using the distributive property:

**Left-hand side (LHS):**

(a-b) - c = a - b - c

**Right-hand side (RHS):**

(a-c) - (b-c) = a - c - b + c

Notice how the 'c' terms cancel out on the RHS:

a - c - b + c = a - b

### Equality Proven

We see that both the LHS and RHS simplify to a - b. This proves that the equation (a-b)-c = (a-c)-(b-c) is indeed a valid identity.

### Applying the Identity

This identity can be helpful in simplifying expressions, especially when dealing with multiple parentheses. By rearranging terms and applying this identity, you can often achieve a more simplified form of the expression.

**Example:**

Simplify the expression: (x - 2) - (3 - x)

Using the identity, we can rewrite this as:

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

### Key Takeaway

The identity (a-b)-c = (a-c)-(b-c) is a valuable tool for simplifying expressions. By understanding the order of operations and applying the distributive property, you can effectively manipulate parentheses and arrive at a more compact form of your equation.