91Ó°ÊÓ

The distributive property is a crucial concept in algebra that allows us to simplify expressions and solve equations efficiently.

This property states that multiplying a number by a sum is the same as doing each multiplication separately and then adding the results. In simpler terms, if you have an equation like \(a(b+c)\), it can be expanded and simplified to \(ab + ac\).

For the given exercise, this concept was applied in the first step to expand both sides of the equation:

• \(3(4b - 7)\) is distributed to become \(12b - 21\).
• \(6(2b + 5)\) is distributed to become \(12b + 30\).

By applying this property, the equation becomes easier to work with. Memorizing this property will help you break down complex algebraic expressions into more manageable parts.

The distributive property is foundational in algebra, playing a significant role in solving equations and simplifying expressions.

Once the distributive property has been used, the next step in simplifying an equation is often to combine like terms. Like terms are terms within an equation that have the same variable raised to the same power or are constant numbers.

In our equation, after using the distributive property, we ended up with:

• \(12b - 21 + 8 = 12b + 30\)

In this equation:

• The terms \(-21\) and \(+8\) are like terms because they are both constants (they don’t have a variable), so they can be combined to \(-13\).

The equation simplifies to \(12b - 13 = 12b + 30\).

Combining like terms helps to simplify equations, allowing us to see the relationships between variables and constants clearly. It's a critical skill for solving algebra problems effectively.

An interesting outcome in solving algebraic equations is encountering a situation where the equation has no solution. This occurs when, after simplification, we end up with a false statement, such as

• \(-13 = 30\)

which can never be true.

In the context of our equation, after isolating the variable \(b\) by subtracting \(12b\) from both sides, we ended up with

• \(-13 = 30\).

Such an equation indicates there is no possible value of \(b\) that would satisfy it.

No solution situations often arise when terms cancel each other out completely, or when the simplification leads to an impossible equation like this one. Recognizing these scenarios is important because it tells us that the given equation isn’t solvable under normal conditions.

• This insight helps us to check for errors if such outcomes appear unexpectedly during problem-solving.

Understanding no solution scenarios is crucial for mastering algebraic problem-solving.