Problem 13 - Prealgebra

Question

Solve each equation. Check your solution. $$-42=\frac{x}{-2}$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 84\).

Step 1: Understand the Equation

We are given the equation \(-42=\frac{x}{-2}\). This equation involves isolating \(x\) which is currently divided by \(-2\). Our goal is to solve for \(x\) by performing operations that will isolate it on one side of the equation.

Step 2: Eliminate the Denominator

To isolate \(x\), we need to eliminate the fraction on the right side by multiplying both sides of the equation by \(-2\), which is the denominator. This gives us: \(-42 \times (-2) = x\).

Step 3: Simplify Both Sides

Multiply \(-42\) by \(-2\) to simplify the left side: \(-42 \times (-2) = 84\). Thus, we now have \(x = 84\).

Step 4: Verify the Solution

To ensure our solution is correct, substitute \(x=84\) back into the original equation: \(-42 = \frac{84}{-2}\). Simplifying the right side gives \(\frac{84}{-2} = -42\). Since the original equation holds true, our solution is verified.

Key Concepts

Checking Solutions

Checking solutions is a crucial step in verifying the accuracy of your answer when solving linear equations. Once you've found a solution, like in our example where we determined that \(x = 84\), it's important to ensure it actually satisfies the original equation. By substituting the solution back into the original equation, you can verify its validity.

• Revisit the original equation: \(-42 = \frac{x}{-2}\).
• Replace \(x\) with your solution: \(-42 = \frac{84}{-2}\).
• Simplify the expression on the right side. Calculate \(\frac{84}{-2}\) which simplifies to \(-42\).

The equation holds true, confirming the solution \(x = 84\) is correct. This process helps identify errors, ensuring that mistakes are fixed before they lead to incorrect conclusions.

Eliminating Fractions

Fractions are often seen as intimidating within equations, but eliminating them is a straightforward process. In our example, we dealt with the fraction \(\frac{x}{-2}\). Here's how we tackled it:

• Identify the fraction in the equation: in \(-42 = \frac{x}{-2}\), \(\frac{x}{-2}\) is the fraction to eliminate.
• Multiply every term by the fraction's denominator: multiplying both sides by \(-2\). This operation aims to "balance" the equation.
• This step changes the equation to: \(-42 \times (-2) = x\).

By eliminating the fraction, we're left with a simple linear equation where \(x\) is isolated on one side, making it easier to solve. The process removes any complication that fractions might introduce, providing clarity and allowing you to focus on isolating the variable.

Isolating Variables

Isolating the variable is a key step in solving linear equations. It involves manipulating the equation so that the variable of interest appears by itself on one side. Let's dive deeper into how this was achieved in our example:

• Start with the altered equation: after multiplying to eliminate fractions, we have \(-42 \times (-2) = x\).
• Simplify the left-hand side: calculate \(-42 \times (-2)\) which results in \(84\).
• Once simplified, your equation is \(x = 84\).

Isolating the variable in this manner allows for a clear identification of its value. It's a systematic approach that involves strategic operations like addition, subtraction, multiplication, or division aimed at leaving the variable alone. Ensuring the variable is isolated means you've logically derived its value based on the premise established by the equation. This provides both clarity and accuracy.