91Ó°ÊÓ

The distributive property is a fundamental algebraic principle used to simplify expressions. It states that for any numbers or variables a, b, and c, the expression a(b + c) is equivalent to ab + ac.

In the original exercise, this property is used to break down the equation: 6(2x + 8) = 4(3x - 6).

We apply the distributive property to expand both sides of the equation:

- Left side: 6(2x + 8) = 6 * 2x + 6 * 8 = 12x + 48
- Right side: 4(3x - 6) = 4 * 3x + 4 * (-6) = 12x - 24

These steps spread out the multiplication over addition or subtraction within parentheses, making the equation simpler to work with. Once expanded, the equation becomes 12x + 48 = 12x - 24.

Simplifying an equation involves combining like terms and making the equation easier to solve. After applying the distributive property to our equation and expanding both sides, we get: 12x + 48 = 12x - 24.

At this point, we simplify by eliminating the same terms from both sides. To do this, we subtract 12x from both sides:
12x + 48 - 12x = 12x - 24 - 12x.

This simplifies to: 48 = -24.

Simplifying equations helps in finding solutions or identifying cases where no solution exists. In situations where identical terms cancel each other out completely, it might indicate special cases like no solution or infinitely many solutions.

Sometimes, during the process of solving an equation, you might find that there is no possible value of the variable that satisfies the equation. Such equations are known as 'no solution' equations.

In our given problem, after simplifying the equation, we obtained: 48 = -24.

This statement is clearly false because 48 can never equal -24. Therefore, we conclude that there is no solution. This indicates that no matter what value of x we try, the original equation can never be satisfied.

When confronted with a result that is a false statement (like 48 = -24), it tells us that the equation has no solution. It's important to carefully check your steps in such cases to ensure that you've not made any mistakes during the simplification process.