Jada Solved The Equation − 4 9 = X 108 -\frac{4}{9}=\frac{x}{108} − 9 4 ​ = 108 X ​ For X X X Using The Steps Below. What Was Jada's Error?$[ \begin{aligned} -\frac{4}{9} & = \frac{x}{108} \ -\frac{4}{9} \left(-\frac{9}{4}\right) & = \frac{x}{108}

Introduction

Solving equations is a fundamental concept in mathematics, and it requires careful attention to detail. In this article, we will analyze the steps taken by Jada to solve the equation $-\frac{4}{9}=\frac{x}{108}$ for $x$. We will identify Jada's error and provide a step-by-step solution to the equation.

Jada's Steps

Jada's steps to solve the equation are as follows:

1. $-\frac{4}{9} = \frac{x}{108}$
2. $-\frac{4}{9} \left(-\frac{9}{4}\right) = \frac{x}{108}$

Jada's Error

Jada's error lies in the second step. When multiplying both sides of the equation by $-\frac{9}{4}$, Jada incorrectly assumed that the negative sign would cancel out. However, the negative sign is not a number that can be multiplied by another number to cancel it out. Instead, the negative sign is an operator that indicates the direction of the number.

Correcting Jada's Error

To correct Jada's error, we need to multiply both sides of the equation by $-\frac{9}{4}$, but we also need to consider the direction of the number. When multiplying a negative number by a positive number, the result is a negative number. Therefore, the correct step should be:

$-\frac{4}{9} \left(-\frac{9}{4}\right) = \frac{x}{108}$

becomes

$\frac{4}{9} \left(\frac{9}{4}\right) = \frac{x}{108}$

Simplifying the Equation

Now that we have corrected Jada's error, we can simplify the equation by multiplying the fractions on the left-hand side:

$\frac{4}{9} \left(\frac{9}{4}\right) = \frac{x}{108}$

becomes

$1 = \frac{x}{108}$

Solving for $x$

To solve for $x$, we can multiply both sides of the equation by $108$:

$1 = \frac{x}{108}$

becomes

$108 = x$

Conclusion

In conclusion, Jada's error was in the second step, where she incorrectly assumed that the negative sign would cancel out. By correcting this error and simplifying the equation, we were able to solve for $x$ and find that $x = 108$.

Discussion

This problem highlights the importance of careful attention to detail when solving equations. It also demonstrates the need to consider the direction of numbers when multiplying them. By following the correct steps and simplifying the equation, we were able to find the correct solution to the equation.

Additional Examples

Here are a few additional examples of equations that require careful attention to detail:

• $-\frac{2}{3} = \frac{x}{12}$
• $\frac{3}{4} = \frac{x}{-16}$
• $-\frac{5}{6} = \frac{x}{24}$

These examples demonstrate the importance of following the correct steps and considering the direction of numbers when solving equations.

Conclusion

Q: What is the main error in Jada's solution?

A: The main error in Jada's solution is in the second step, where she incorrectly assumed that the negative sign would cancel out when multiplying both sides of the equation by $-\frac{9}{4}$.

Q: Why is it incorrect to assume that the negative sign will cancel out?

A: The negative sign is an operator that indicates the direction of the number. When multiplying a negative number by a positive number, the result is a negative number. Therefore, the negative sign does not cancel out, but rather changes the direction of the number.

Q: How can we correct Jada's error?

A: To correct Jada's error, we need to multiply both sides of the equation by $-\frac{9}{4}$, but we also need to consider the direction of the number. When multiplying a negative number by a positive number, the result is a negative number. Therefore, the correct step should be:

$\frac{4}{9} \left(\frac{9}{4}\right) = \frac{x}{108}$

Q: What is the correct solution to the equation?

A: The correct solution to the equation is $x = 108$. To solve for $x$, we can multiply both sides of the equation by $108$:

$1 = \frac{x}{108}$

becomes

$108 = x$

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

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

• Incorrectly assuming that the negative sign will cancel out
• Failing to consider the direction of numbers when multiplying them
• Not following the correct steps to simplify the equation
• Not checking the solution for errors

Q: How can we ensure that we are solving equations correctly?

A: To ensure that we are solving equations correctly, we should:

• Carefully read and understand the equation
• Follow the correct steps to simplify the equation
• Consider the direction of numbers when multiplying them
• Check the solution for errors
• Double-check our work to ensure that we have not made any mistakes

Q: What are some additional tips for solving equations?

A: Some additional tips for solving equations include:

• Using a systematic approach to solve the equation
• Checking the solution for errors
• Using a calculator or computer program to check the solution
• Asking for help if you are unsure about the solution
• Practicing solving equations to build your skills and confidence

Q: How can we apply the concepts learned in this article to real-world problems?

A: The concepts learned in this article can be applied to real-world problems in a variety of ways, including:

• Solving financial problems, such as calculating interest rates or investment returns
• Solving scientific problems, such as calculating the trajectory of a projectile or the motion of an object
• Solving engineering problems, such as designing a bridge or a building
• Solving business problems, such as calculating profits or losses

By applying the concepts learned in this article, we can develop our problem-solving skills and become more confident and proficient in solving equations.