Problem 66 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ \frac{x}{-4}=-3 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 12\).

Step 1: Identify the Equation

The given equation is \( \frac{x}{-4} = -3 \). The goal is to solve for the variable \( x \).

Step 2: Multiply Both Sides by -4

To isolate \( x \), multiply both sides of the equation by \(-4\). This will eliminate the fraction:\[-4 \times \frac{x}{-4} = -4 \times (-3) \]

Step 3: Simplify the Equation

When you multiply \(-4\) with \(\frac{x}{-4}\), \(x\) is left alone on the left side:\[x = 12\]This is because multiplying by \(-4\) cancels out the denominator \(-4\) on the left, and on the right side, \(-4\) multiplied by \(-3\) is \(12\).

Step 4: Verify the Solution

Substitute \(x = 12\) back into the original equation to ensure it satisfies the equation:\[\frac{12}{-4} = -3\]The equation holds true, confirming that \(x = 12\) is indeed the correct solution.

Key Concepts

Solving Linear Equations

Solving linear equations is a foundational skill in algebra. Linear equations involve an equal sign and typically a single variable, such as "x," which we aim to solve for. The process often involves isolating this variable on one side of the equation. This might initially sound complex, but it's all about performing operations that keep the equation balanced. For instance, consider the equation \( \frac{x}{-4} = -3 \). Here the goal is to solve for "x" by removing the fraction. This is typically done by reversing operations (like division) to isolate "x." In this example, we multiplied both sides by -4. This operation cancels out -4 on the left side and leaves \( x = 12 \) on the right side. This process shows how reversible operations help us solve linear equations effectively.

Operations with Integers

Understanding operations with integers is crucial in solving algebraic problems. Integers are whole numbers that can be positive, negative, or zero. When manipulating equations, it's important to apply the correct operations to maintain equality. For example, when solving \( \frac{x}{-4} = -3 \), you need to multiply both sides by -4.

• This multiplication is crucial to eliminate the fraction and isolate "x."
• Performing \(-4 \times -3\) gives us 12, highlighting the rule that the product of two negative numbers is positive.

Being comfortable with multiplication and division of integers can simplify the solution of equations by making it easier to isolate variables.

Equation Verification

Equation verification is the final step in solving equations, ensuring our solution satisfies the original equation. This is a critical process to confirm accuracy. After solving an equation, such as finding \( x = 12 \) in \( \frac{x}{-4} = -3 \), it's important to substitute the solution back into the original equation to check if it holds true.

• Replace "x" with 12: \( \frac{12}{-4} = -3 \).
• If both sides of the equation match, the solution is verified and correct.

Verification provides confidence that we've accurately solved the equation and serves as a good practice to catch any possible calculation errors. Always remember this final step to ensure your solution is indeed valid.