Problem 14 - Algebra A Combined Function

Question

Solve each equation. Check each solution. See Examples 1 through \(6 .\) \(\frac{f}{-5}=0\)

Step-by-Step Solution

Answer

The solution is \( f = 0 \). Substituting back shows the solution is correct.

Step 1: Understanding the Equation

We are given the equation \( \frac{f}{-5} = 0 \). This equation suggests that when \( f \) is divided by -5, the result is 0.

Step 2: Isolating the Variable

To solve for \( f \), we multiply both sides of the equation by -5 to eliminate the denominator: \( f = 0 \times (-5) \).

Step 3: Simplifying the Equation

Perform the multiplication: \( f = 0 \times (-5) = 0 \). So, \( f = 0 \).

Step 4: Checking the Solution

Substitute \( f = 0 \) back into the original equation to verify: \( \frac{0}{-5} = 0 \). The equation holds true, confirming that the solution is correct.

Key Concepts

Isolating the Variable

When solving linear equations, one of the first steps is often "isolating the variable." This means you want to get the variable by itself on one side of the equation to see what it equals. Think of it as solving a mystery where the variable is the missing piece of information you need to find.
In the equation \( \frac{f}{-5} = 0 \), the variable \( f \) is currently divided by -5. To isolate \( f \), you need to undo this division by multiplying both sides by -5. This is similar to using a key to unlock a lock:

• Multiply both sides by -5, which cancels out the division on the left side.
• This leaves you with \( f = 0 \times (-5) \).

After multiplying, you're left with a simple equation: \( f = 0 \). Now, the variable \( f \) is isolated and you know its value.

Checking Solutions

After solving for the variable, it's crucial to "check the solution." This step ensures that your solution makes the original equation valid. It acts as a verification process—a big thumbs-up to tell you that you have reasoned correctly. Here’s how checking solutions should be done:
Return to the original equation, \( \frac{f}{-5} = 0 \), and substitute the value you found for the variable. So, substitute \( f = 0 \):

• The left side becomes \( \frac{0}{-5} \).
• This simplifies to 0, which matches the right side of the equation.

Since both sides of the equation are equal after substitution, your solution \( f = 0 \) is correct. Checking might seem tedious, but it confirms your solution is not a fluke and adheres to the rules of algebra.

Simplification Steps

When solving equations, "simplification steps" are the acts of reducing the equation to its simplest form until the answer reveals itself. Each simplification makes the equation a little less complex, allowing us to focus on what really matters—the solution. In our problem, simplifying involved several tasks:
1. **Multiply Zero:** Multiplying 0 by any number is crucial. It's zero and incredibly simple yet powerful. In this equation, \( 0 \times (-5) \) naturally becomes 0.
2. **Simplifying Expressions:** Even though this equation was straightforward, knowing how to handle expressions like \( \frac{f}{-5} = 0 \) is key. It requires recognizing that \( f \) is alone on one side, making comparison possible.

• If more complex expressions were present, handle them one step at a time.

By adhering to these steps, you ensure each piece of the puzzle logically follows from the last.