91Ó°ÊÓ

The distributive property is essential when solving equations, especially when dealing with parentheses. In our exercise, the equation \(-2.5x + 5.76 = 0.4(6 - 5x)\) requires us to distribute 0.4 across the terms inside the parentheses. This means we need to multiply 0.4 by each term within the parentheses separately.

Applying the distributive property, we have:
- \(0.4 * 6 = 2.4\)
- \(0.4 * (-5x) = -2x\)

Once distributed, the equation becomes:
\(-2.5x + 5.76 = 2.4 - 2x\).

Breaking down the problem this way simplifies the equation and eliminates the parentheses, making it easier to work with.

After distributing, the next step is to isolate the variable, making it easier to solve for \(x\).

First, we need to collect all the \(x\)-terms on one side of the equation. To achieve this, we add \(2x\) to both sides, canceling out \(x\) on the right side:
\(-2.5x + 2x + 5.76 = 2.4 - 2x + 2x\)

This simplifies to:
\(-0.5x + 5.76 = 2.4\)

Now, we need to isolate \(x\) completely. Next, subtract \(5.76\) from both sides:
\(-0.5x + 5.76 - 5.76 = 2.4 - 5.76\)

This reduces to:
\(-0.5x = -3.36\)

By carefully isolating the variable, solving the equation becomes much more straightforward.

Balancing equations means ensuring both sides of the equation remain equal as you perform operations. Every step should preserve the balance.

In our problem, we first distributed 0.4 across the parentheses to have both sides balanced:
\(-2.5x + 5.76 = 2.4 - 2x\)

Then, we added \(2x\) to both sides to collect the \(x\)-terms:
\(-2.5x + 2x + 5.76 = 2.4 - 2x + 2x\)

This balanced step-by-step approach led us to:
\(-0.5x + 5.76 = 2.4\)

Finally, solving for \(x\) involved dividing both sides by \(-0.5\) after isolating \(x\) on one side:
\(\frac{-0.5x}{-0.5} = \frac{-3.36}{-0.5}\)
\(x = 6.72\)

Each operation maintained the equation's balance, ensuring the equality holds true.