Analyze The Following Statements Based On The Given Solution:Given Solution:${ \begin{aligned} \frac{5}{12} & =-\frac{x}{420} \ \frac{5}{12}(420) & =-\frac{x}{420}(420) \ x & =175 \end{aligned} }$Determine The Correctness Of The Following

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Introduction

In mathematics, solving equations is a fundamental concept that requires a deep understanding of algebraic manipulations. The given solution involves a simple equation with a variable, and our task is to analyze its correctness. We will break down the solution step by step, highlighting the key operations and identifying any potential errors.

The Given Solution

The given solution is as follows:

512=−x420512(420)=−x420(420)x=175\begin{aligned} \frac{5}{12} & =-\frac{x}{420} \\ \frac{5}{12}(420) & =-\frac{x}{420}(420) \\ x & =175 \end{aligned}

Step 1: Multiplying Both Sides by 420

The first step involves multiplying both sides of the equation by 420. This operation is valid since 420 is a non-zero constant.

512(420)=−x420(420)\frac{5}{12}(420) = -\frac{x}{420}(420)

Step 2: Simplifying the Equation

Multiplying both sides by 420 results in:

5×42012=−x×420\frac{5 \times 420}{12} = -x \times 420

Simplifying the left-hand side, we get:

210012=−x×420\frac{2100}{12} = -x \times 420

Further simplifying, we have:

175=−x×420175 = -x \times 420

Step 3: Solving for x

Now, we need to isolate the variable x. To do this, we divide both sides of the equation by -420.

175−420=x\frac{175}{-420} = x

Simplifying the fraction, we get:

−175420=x-\frac{175}{420} = x

Conclusion

Based on the step-by-step analysis, we can conclude that the given solution is INCORRECT. The correct solution involves isolating the variable x by dividing both sides of the equation by -420, resulting in:

x=−175420x = -\frac{175}{420}

This value is different from the given solution, which states that x = 175. Therefore, the given solution is incorrect.

Discussion

The given solution involves a simple equation with a variable, and our task is to analyze its correctness. We broke down the solution step by step, highlighting the key operations and identifying any potential errors. The correct solution involves isolating the variable x by dividing both sides of the equation by -420, resulting in a different value.

Key Takeaways

  • The given solution is incorrect.
  • The correct solution involves isolating the variable x by dividing both sides of the equation by -420.
  • The value of x is -175/420, which is different from the given solution.

Recommendations

  • When solving equations, it is essential to follow the correct order of operations and to isolate the variable correctly.
  • Always check the solution by plugging it back into the original equation to ensure its correctness.
  • Practice solving equations with variables to develop a deep understanding of algebraic manipulations.
    Frequently Asked Questions (FAQs) About the Given Solution ================================================================

Q: What is the given solution trying to solve?

A: The given solution is trying to solve the equation 512=−x420\frac{5}{12} = -\frac{x}{420} for the variable x.

Q: Is the given solution correct?

A: No, the given solution is incorrect. The correct solution involves isolating the variable x by dividing both sides of the equation by -420, resulting in a different value.

Q: What is the correct solution to the equation?

A: The correct solution to the equation is x=−175420x = -\frac{175}{420}.

Q: Why is the given solution incorrect?

A: The given solution is incorrect because it does not follow the correct order of operations and does not isolate the variable x correctly.

Q: What are some common mistakes to avoid when solving equations?

A: Some common mistakes to avoid when solving equations include:

  • Not following the correct order of operations
  • Not isolating the variable correctly
  • Not checking the solution by plugging it back into the original equation
  • Not practicing solving equations with variables to develop a deep understanding of algebraic manipulations

Q: How can I improve my skills in solving equations?

A: To improve your skills in solving equations, you can:

  • Practice solving equations with variables regularly
  • Review the correct order of operations and how to isolate variables correctly
  • Check your solutions by plugging them back into the original equation
  • Seek help from a teacher or tutor if you are struggling with a particular concept

Q: What are some real-world applications of solving equations?

A: Solving equations has many real-world applications, including:

  • Physics and engineering: Solving equations is essential in physics and engineering to model and analyze complex systems.
  • Economics: Solving equations is used in economics to model and analyze economic systems.
  • Computer science: Solving equations is used in computer science to develop algorithms and solve problems.
  • Data analysis: Solving equations is used in data analysis to model and analyze complex data sets.

Q: Can you provide more examples of solving equations?

A: Yes, here are a few more examples of solving equations:

  • 23=x9\frac{2}{3} = \frac{x}{9}: Solve for x.
  • x+5=11x + 5 = 11: Solve for x.
  • x2+3=7\frac{x}{2} + 3 = 7: Solve for x.

These are just a few examples, and there are many more equations that can be solved using algebraic manipulations.