91Ó°ÊÓ

The distributive property is a fundamental concept in algebra used to simplify expressions. When the distributive property is applied, it involves multiplying a single term by each term within a parenthesis, thus "distributing" the multiplication across all terms. This rule is written in general terms as:

\[a(b + c) = ab + ac\]

In the exercise, the expression \(-0.24(C-0.50)\) needed to be simplified first. Using the distributive property, we multiply \(-0.24\) by both \(C\) and \(-0.50\), resulting in \(-0.24C + 0.12\). This step is crucial in setting up the equation for easier solving in the next steps. Always double-check your signs (positive/negative) when distributing; incorrect signs can lead to wrong solutions.

Isolating variables is the process of manipulating an equation in such a way that you have the unknown term, like \(C\), on one side by itself. This is important to solve equations for the variable in question.

In our example, after distributing, we have the equation \(-0.24C + 0.12 = 0.63\). Our goal is to isolate the term with \(C\). To do this, we perform the inverse operation of addition, which is subtraction.

• Subtract \(0.12\) from both sides: \(-0.24C + 0.12 - 0.12 = 0.63 - 0.12\)
• This simplifies to \(-0.24C = 0.51\)

By following these steps, we effectively isolate the variable term, making it easier to solve the equation for \(C\).

Solving linear equations involves finding the value of the unknowns that satisfy the equation. Once the variable term is isolated, like in \(-0.24C = 0.51\), you solve for the variable by performing inverse operations.

To find \(C\), you need to apply the inverse operation of multiplication, which is division. Here are the steps:

• Divide both sides of the equation by the coefficient of \(C\), which is \(-0.24\).
• This results in \[C = \frac{0.51}{-0.24}\]
• Finally, compute the division to get \(C \approx -2.125\).

Remember, accurate calculations are important, so handling decimals and negative signs with care is essential for finding the correct solution.