Problem 107 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ \frac{x}{-3}=8 $$

Step-by-Step Solution

Answer

The solution is \( x = -24 \).

Step 1: Identify the equation

The given equation is \( \frac{x}{-3} = 8 \). Our task is to solve for \( x \).

Step 2: Isolate x

To isolate \( x \), we need to eliminate \(-3\) from the denominator. Multiply both sides of the equation by \(-3\).

Step 3: Perform the multiplication

When we multiply both sides by \(-3\), we get:\[ x = 8 \times (-3) \]

Step 4: Simplify the expression

Calculate \( 8 \times (-3) \) to get: \[ x = -24 \]

Step 5: Verify the solution

To verify, substitute \( x = -24 \) back into the original equation: \( \frac{-24}{-3} = 8 \). Simplifying the left side gives \( 8 = 8 \), which confirms our solution is correct.

Key Concepts

Equation Solving

Solving equations is a fundamental skill in mathematics. It involves finding the value of the unknown variable that makes the equation true. The goal is to manipulate the equation in such a way that the variable is alone on one side, revealing what it equals. This process is crucial because it helps us understand and model real-world scenarios numerically. For instance, in our given problem \( \frac{x}{-3} = 8 \), the task is to determine the value of \( x \) that satisfies this equation.
To solve an equation correctly, follow these simple steps:

• Identify the equation and the variable you need to find.
• Use operations like addition, subtraction, multiplication, or division to isolate the variable.
• Simplify both sides whenever necessary.
• Always double-check your work by verifying the solution.

Each of these steps contributes to an organized approach toward solving equations, ensuring accuracy and efficiency.

Isolating Variables

Isolating the variable is a key strategy in solving equations. It means rearranging the equation so that the variable you want to solve for is by itself on one side of the equation. This process helps clarify what the variable equals, making it easy to plug this value back into practical or theoretical contexts.
In our problem, the equation initially is \( \frac{x}{-3} = 8 \). To isolate \( x \), we need to get rid of the \(-3\) that is dividing \( x \). We do this by performing the opposite operation of division, which is multiplication:

• Multiply both sides of the equation by \(-3\).
• This operation leaves \( x \) alone on one side: \( x = 8 \times (-3) \).

By isolating \( x \), you directly translate the relationship defined by the equation, allowing easier computation of the solution.

Verifying Solutions

Verifying solutions is an essential step to ensure that you've solved the equation correctly. It involves substituting the solution back into the original equation to check that it holds true. This step serves as a confirmation and helps avoid errors.
For the equation \( \frac{x}{-3} = 8 \), once we found \( x = -24 \), we verify the solution by plugging it back into the original equation:

• Replace \( x \) in the equation with \(-24\).
• This yields \( \frac{-24}{-3} = 8 \).
• Simplifying the left side, \( -24 \div -3 = 8 \), confirms that the left side equals the right side, verifying our solution.

Verification is like a safety net catching mistakes before you finalize your answer, ensuring reliability in your mathematical work.

Simplification in Mathematics

Simplification in mathematics refers to reducing expressions to their simplest form, while maintaining their original value or meaning. This process is fundamental when working with equations, as it makes them easier to understand and solve.
In our problem, simplifying the expression was necessary to find the value of \( x \). After isolating \( x \), we performed the multiplication:

• Calculate \( 8 \times (-3) \), which simplifies to \(-24\).
• This simplification reveals the solution: \( x = -24 \).

Without simplification, the process of solving and verifying solutions would be cumbersome. Simplifying expressions not only aids in clarity, but also in verifying the accuracy of calculations, ensuring each step logically follows the last.