91Ó°ÊÓ

The distributive property is a fundamental concept in algebra that allows us to simplify expressions by removing parentheses. It states that for any numbers or expressions, \( a(b + c) = ab + ac \). This property helps in breaking down complex expressions into simpler parts.

In the given problem, we have an expression \(-2(3x - 4)\). Here, the number \(-2\) multiplies the terms inside the parentheses. Applying the distributive property means:

• Multiplying \(-2\) by \(3x\) gives us \(-6x\)
• Multiplying \(-2\) by \(-4\) results in \(+8\)

After distributing, the expression inside the parentheses becomes \(-6x + 8\).

This step is crucial because it sets the stage for further simplification. By understanding and applying the distributive property, we can rearrange expressions and make them easier to work with.

Combining like terms is a method used to simplify expressions by merging similar kinds of terms. "Like terms" are terms that have the same variable raised to the same power. Only the coefficients of the terms are different.

For instance, in our expression \(-6x + 8 + 7x - 6\), the like terms are:

• Variable terms: \(-6x\) and \(7x\)
• Constant terms: \(+8\) and \(-6\)

To combine the variable terms, simply add their coefficients: \(-6 + 7 = 1\) which gives us the term \(x\).

For the constants, calculate \(8 - 6 = 2\). This makes the expression simpler: \(x + 2\).

Combining like terms helps reduce the expression to its simplest form, making it more manageable and clearer.

Simplifying algebraic expressions involves using various algebraic techniques to transform a complex expression into its simplest form. This process makes calculations easier and expressions clearer.

Let's summarize the simplification of the original problem \(-2(3x - 4) + 7x - 6\):

• First, use the distributive property: this transforms the expression to \(-6x + 8\).
• Next, substitute this result back into the entire expression, resulting in \(-6x + 8 + 7x - 6\).
• Finally, combine like terms: adding the \(x\) terms results in \(x\) and simplifying the constants \(8 - 6\) results in \(2\).

Thus, the expression simplifies to \(x + 2\).

This entire process showcases how applying mathematical principles like the distributive property and combining like terms can simplify expressions effectively. It enhances one's problem-solving skills and provides a finer understanding of algebraic manipulations.