Solve The Equation: $ \frac{84}{12} = \frac{28}{x} }$Please Select The Best Answer From The Choices Provided A. { X = 4 $ $B. { X = 28 $}$C. { X = 12 $}$D. { X = 7 $}$

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Introduction

In mathematics, equations are a fundamental concept that helps us solve problems and understand various mathematical relationships. One of the most common types of equations is the proportionality equation, which states that two ratios are equal. In this article, we will focus on solving a specific proportionality equation, which is 8412=28x\frac{84}{12} = \frac{28}{x}. We will break down the solution step by step and provide a clear explanation of each step.

Understanding the Equation

The given equation is a proportionality equation, which means that the two ratios are equal. The equation is 8412=28x\frac{84}{12} = \frac{28}{x}. To solve this equation, we need to find the value of xx that makes the two ratios equal.

Step 1: Simplify the Left-Hand Side

The first step in solving the equation is to simplify the left-hand side. We can do this by dividing both the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD of 84 and 12 is 12. Therefore, we can simplify the left-hand side as follows:

8412=71\frac{84}{12} = \frac{7}{1}

Step 2: Cross-Multiply

The next step is to cross-multiply the two ratios. This means that we multiply the numerator of the left-hand side by the denominator of the right-hand side, and vice versa. In this case, we get:

7x=28×127x = 28 \times 12

Step 3: Simplify the Right-Hand Side

The next step is to simplify the right-hand side. We can do this by multiplying 28 and 12:

7x=3367x = 336

Step 4: Solve for x

The final step is to solve for xx. We can do this by dividing both sides of the equation by 7:

x=3367x = \frac{336}{7}

Step 5: Simplify the Result

The final step is to simplify the result. We can do this by dividing 336 by 7:

x=48x = 48

Conclusion

In conclusion, the value of xx that makes the two ratios equal is 48. This is the solution to the equation 8412=28x\frac{84}{12} = \frac{28}{x}. We hope that this step-by-step guide has helped you understand how to solve proportionality equations.

Answer

The correct answer is:

A. x=48x = 48

Discussion

This equation is a classic example of a proportionality equation. The key concept here is that the two ratios are equal, and we need to find the value of xx that makes this true. The solution involves simplifying the left-hand side, cross-multiplying, simplifying the right-hand side, and finally solving for xx. We hope that this discussion has helped you understand the concept of proportionality equations and how to solve them.

Related Topics

  • Proportionality equations
  • Ratios
  • Simplifying fractions
  • Cross-multiplication
  • Solving for variables

Practice Problems

  • Solve the equation 1520=3x\frac{15}{20} = \frac{3}{x}
  • Solve the equation 2436=4x\frac{24}{36} = \frac{4}{x}
  • Solve the equation 912=3x\frac{9}{12} = \frac{3}{x}

Conclusion

Q: What is a proportionality equation?

A: A proportionality equation is a mathematical equation that states that two ratios are equal. It is a fundamental concept in mathematics that helps us understand various mathematical relationships.

Q: How do I simplify a proportionality equation?

A: To simplify a proportionality equation, you need to follow these steps:

  1. Simplify the left-hand side by dividing both the numerator and the denominator by their greatest common divisor (GCD).
  2. Cross-multiply the two ratios.
  3. Simplify the right-hand side by multiplying the two numbers.
  4. Solve for the variable by dividing both sides of the equation by the coefficient of the variable.

Q: What is cross-multiplication?

A: Cross-multiplication is a technique used to solve proportionality equations. It involves multiplying the numerator of the left-hand side by the denominator of the right-hand side, and vice versa.

Q: How do I solve for x in a proportionality equation?

A: To solve for x in a proportionality equation, you need to follow these steps:

  1. Simplify the left-hand side by dividing both the numerator and the denominator by their greatest common divisor (GCD).
  2. Cross-multiply the two ratios.
  3. Simplify the right-hand side by multiplying the two numbers.
  4. Solve for x by dividing both sides of the equation by the coefficient of x.

Q: What is the difference between a proportionality equation and an algebraic equation?

A: A proportionality equation is a mathematical equation that states that two ratios are equal, while an algebraic equation is a mathematical equation that involves variables and constants. Proportionality equations are a type of algebraic equation.

Q: Can I use a calculator to solve proportionality equations?

A: Yes, you can use a calculator to solve proportionality equations. However, it is always a good idea to check your work by hand to ensure that you have made no mistakes.

Q: How do I check my work when solving a proportionality equation?

A: To check your work when solving a proportionality equation, you need to follow these steps:

  1. Simplify the left-hand side by dividing both the numerator and the denominator by their greatest common divisor (GCD).
  2. Cross-multiply the two ratios.
  3. Simplify the right-hand side by multiplying the two numbers.
  4. Solve for x by dividing both sides of the equation by the coefficient of x.
  5. Check that your answer is consistent with the original equation.

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

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

  • Not simplifying the left-hand side by dividing both the numerator and the denominator by their greatest common divisor (GCD).
  • Not cross-multiplying the two ratios.
  • Not simplifying the right-hand side by multiplying the two numbers.
  • Not solving for x by dividing both sides of the equation by the coefficient of x.
  • Not checking your work by hand to ensure that you have made no mistakes.

Q: Can I use a graphing calculator to solve proportionality equations?

A: Yes, you can use a graphing calculator to solve proportionality equations. However, it is always a good idea to check your work by hand to ensure that you have made no mistakes.

Q: How do I use a graphing calculator to solve proportionality equations?

A: To use a graphing calculator to solve proportionality equations, you need to follow these steps:

  1. Enter the equation into the calculator.
  2. Use the calculator to graph the equation.
  3. Use the calculator to find the x-intercept of the graph.
  4. Check that your answer is consistent with the original equation.

Conclusion

In conclusion, solving proportionality equations involves simplifying the left-hand side, cross-multiplying, simplifying the right-hand side, and finally solving for x. We hope that this article has provided a clear and concise guide to solving proportionality equations. With practice and patience, you will become proficient in solving these types of equations.