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Chapter 3: Problem 44

Solve and check each equation. $$4 f=0$$

Short Answer

Expert verified
The solution to the given equation \(4f=0\) is \(f=0\). This is verified by substituting the value of f back into the original equation, which resulted in both sides being equal: \(4(0)=0 \Rightarrow 0=0\).

Step by step solution

01

Identify the equation

The given equation is \(4f=0\). Our goal is to solve this equation for f.
02

Isolate the variable f

To isolate f, we need to divide both sides of the equation by 4. This will give us the value of f. \(4f = 0 \Rightarrow \frac{4f}{4} = \frac{0}{4}\)
03

Simplify and find the solution

After dividing both sides by 4, we have: \(f = 0\) So, the solution to the given equation is \(f=0\).
04

Check the solution

To check our solution, substitute the value of f back into the original equation and verify if both sides are equal: \(4(0) = 0\) This simplifies to: \(0 = 0\) Since both sides of the equation are equal, our solution is correct. Therefore, the solution to the given equation \(4f=0\) is \(f=0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Linear Equations
When dealing with linear equations like the one presented in the exercise, our main goal is to find the value of the unknown variable that satisfies the equation. Linear equations typically take the form of \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. In our case, the equation was \( 4f = 0 \).

To solve linear equations, we follow a series of straightforward steps:
  • Identify the structure of the equation to determine the operation needed for simplification.
  • Work through the operations inversely by "undoing" what's been done to the variable. This usually involves basic arithmetic operations.
  • Simplify both sides of the equation to find the unknown variable's value.
Solving linear equations helps develop problem-solving skills and a solid understanding of algebraic manipulations, which are foundational in many advanced math topics.
Variable Isolation
In algebra, isolating the variable is a crucial step when solving equations. This process entails manipulating the equation in such a way that the variable appears alone on one side of the equation. In our problem, we aim to isolate \( f \) in the equation \( 4f = 0 \).

To isolate \( f \), we perform the following:
  • Look for any multiplication or division done to the variable. Here, \( f \) is multiplied by 4.
  • Use inverse operations to counter these effects. Since \( f \) is multiplied by 4, we divide both sides of the equation by 4.
  • Re-evaluate the equation after performing the operation to ensure the variable is isolated.
The result of this operation shows \( f = 0 \). Now, \( f \) stands alone, confirming that it has been isolated correctly. Understanding this concept is vital for efficiently solving equations, as it equips you with the tools to handle more complex algebraic expressions.
Equation Verification
Checking your work is a vital aspect of solving equations, ensuring that your solution is indeed correct. In the verification phase, you substitute your solution back into the original equation to confirm that both sides maintain equality. For the equation \( 4f = 0 \), we found \( f = 0 \).

To verify this, we:
  • Replace the variable \( f \) with the solution found: \( 4(0) = 0 \).
  • Simplify the expression to see if the left side equals the right side: \( 0 = 0 \).
  • Confirm that the equality holds, proving that our solution is correct.
Verification reinforces your understanding and provides confidence in your solution. It also acts as a safeguard against simple arithmetic errors or incorrect algebraic manipulations made during solving. Always verify your solutions as a best practice in mathematics.

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Most popular questions from this chapter

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