91Ó°ÊÓ

When dealing with algebraic expressions, getting the right answer depends greatly on following the correct order of operations. This order is often remembered by the acronym BIDMAS or BODMAS:
- **B**rackets first
- **I**ndices (or **O**rders, such as exponents and roots)
- **D**ivision and **M**ultiplication (from left to right)
- **A**ddition and **S**ubtraction (from left to right)
Starting with parentheses, we simplify everything inside them before moving on to exponents, and so on. This ensures that each part of the equation is handled properly, just like when solving our original exercise.
For example, in the given expression, we need to solve the operations in brackets first: \( (-2+5)(5+(-3)) \). By calculating these, we find \( 3 \) and \( 2 \), respectively, which keep everything in place for subsequent steps.

Substitution is a fundamental part of evaluating algebraic expressions. It involves replacing variables with their given numerical values. This step makes abstract expressions more concrete and manageable.
In our exercise, we substitute the values \( x = -2 \), \( y = 5 \), and \( z = -3 \) directly into the expression. So \( rac{(x+y)(y+z)}{x+y+z} \) becomes \( rac{(-2+5)(5+(-3))}{-2+5+(-3)} \).
This process simplifies the original problem into one with plain numbers, making calculations possible and preparing us for further steps. Substitution is a powerful technique because it transforms expressions, fostering an easier path to the solution.

One of the fundamental rules in mathematics is that division by zero is undefined. Think of division as sharing items among a group of people. If the group is empty (zero people), there’s no valid way to share the items.
In our problem, after substitution and simplification, we reached the expression \( \frac{6}{0} \). Since dividing by zero can’t be done, the expression doesn't have a numerical answer.
- **Why is it undefined?**
- Dividing a number by zero doesn’t provide a result that fits established number systems.
- It would imply something impossible or infinite, which doesn’t mesh with real numbers.
Always remember, if you're ever faced with a zero in the denominator of a fraction or division, the answer is simply 'undefined'!