-c-%3D-(a-c)-(b-c).jpeg "(a-b)-c = (a-c)-(b-c)")
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):
- Parentheses / Brackets
- Exponents / Orders
- Multiplication and Division (from left to right)
- Addition and Subtraction (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.