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) &

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Introduction

Solving equations is a fundamental concept in mathematics, and it's essential to understand the steps involved to arrive at the correct solution. In this article, we will discuss a common mistake made by students when solving equations, using the example of Jada's error in solving the equation −49=x108-\frac{4}{9}=\frac{x}{108}.

The Correct Solution

To solve the equation −49=x108-\frac{4}{9}=\frac{x}{108}, we need to isolate the variable xx. The first step is to multiply both sides of the equation by the reciprocal of the coefficient of xx, which is 1081\frac{108}{1}.

−49=x108-\frac{4}{9}=\frac{x}{108}

−49⋅1081=x108⋅1081-\frac{4}{9}\cdot\frac{108}{1}=\frac{x}{108}\cdot\frac{108}{1}

−4⋅1089⋅1=x⋅108108⋅1-\frac{4\cdot108}{9\cdot1}=\frac{x\cdot108}{108\cdot1}

−4329=108x108-\frac{432}{9}=\frac{108x}{108}

−4329=108x108-\frac{432}{9}=\frac{108x}{108}

Now, we can simplify the equation by dividing both sides by −1-1.

4329=−108x108\frac{432}{9}=-\frac{108x}{108}

4329=−x\frac{432}{9}=-x

4329=−x\frac{432}{9}=-x

To solve for xx, we can multiply both sides of the equation by −1-1.

4329=−x\frac{432}{9}=-x

4329⋅(−1)=x\frac{432}{9}\cdot(-1)=x

−4329=x-\frac{432}{9}=x

−481=x-\frac{48}{1}=x

Therefore, the correct solution to the equation −49=x108-\frac{4}{9}=\frac{x}{108} is x=−48x=-48.

Jada's Error

Jada's error occurred when she multiplied both sides of the equation by −49-\frac{4}{9} instead of 94\frac{9}{4}.

−49=x108-\frac{4}{9}=\frac{x}{108}

−49(−94)=x108(−94)-\frac{4}{9}\left(-\frac{9}{4}\right)=\frac{x}{108}\left(-\frac{9}{4}\right)

49=−9x108\frac{4}{9}=-\frac{9x}{108}

49=−9x108\frac{4}{9}=-\frac{9x}{108}

Jada's mistake was to multiply both sides of the equation by the reciprocal of the coefficient of xx, which is −49-\frac{4}{9}, instead of 94\frac{9}{4}.

Discussion

When solving equations, it's essential to understand the concept of reciprocals and how to use them to isolate the variable. In this case, Jada's error occurred when she multiplied both sides of the equation by the wrong reciprocal.

Conclusion

Solving equations requires attention to detail and a clear understanding of the steps involved. Jada's error in solving the equation −49=x108-\frac{4}{9}=\frac{x}{108} highlights the importance of using the correct reciprocal to isolate the variable. By following the correct steps and using the correct reciprocal, we can arrive at the correct solution to the equation.

Common Mistakes to Avoid

When solving equations, it's essential to avoid common mistakes such as:

  • Multiplying both sides of the equation by the wrong reciprocal
  • Dividing both sides of the equation by zero
  • Forgetting to multiply both sides of the equation by the reciprocal of the coefficient of xx

By understanding these common mistakes and following the correct steps, we can ensure that we arrive at the correct solution to the equation.

Real-World Applications

Solving equations has numerous real-world applications, including:

  • Physics: Solving equations is essential in physics to describe the motion of objects and the forces acting on them.
  • Engineering: Solving equations is used in engineering to design and optimize systems, such as bridges and buildings.
  • Economics: Solving equations is used in economics to model the behavior of economic systems and make predictions about future trends.

Final Thoughts

Introduction

Solving equations is a fundamental concept in mathematics, and it's essential to understand the steps involved to arrive at the correct solution. In this article, we will provide a Q&A guide to help you understand the concept of solving equations and common mistakes to avoid.

Q: What is an equation?

A: An equation is a mathematical statement that expresses the equality of two expressions. It consists of two parts: the left-hand side (LHS) and the right-hand side (RHS).

Q: What is the purpose of solving equations?

A: The purpose of solving equations is to find the value of the variable (or variables) that makes the equation true. Solving equations is essential in mathematics, science, and engineering to describe the behavior of systems and make predictions about future trends.

Q: What are the steps involved in solving equations?

A: The steps involved in solving equations are:

  1. Read and understand the equation
  2. Identify the variable (or variables) to be solved
  3. Use algebraic operations (addition, subtraction, multiplication, and division) to isolate the variable
  4. Check the solution by substituting it back into the original equation

Q: What is the concept of reciprocals in solving equations?

A: In solving equations, reciprocals are used to isolate the variable. A reciprocal is a number that, when multiplied by another number, gives 1. For example, the reciprocal of 2 is 1/2.

Q: What is the difference between a reciprocal and a fraction?

A: A reciprocal is a number that, when multiplied by another number, gives 1. A fraction is a number that represents a part of a whole. For example, 1/2 is a fraction, but 2 is its reciprocal.

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

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

  • Multiplying both sides of the equation by the wrong reciprocal
  • Dividing both sides of the equation by zero
  • Forgetting to multiply both sides of the equation by the reciprocal of the coefficient of x
  • Not checking the solution by substituting it back into the original equation

Q: How can I check my solution to an equation?

A: To check your solution to an equation, substitute the solution back into the original equation and verify that it is true. This is called "back-substitution."

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

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

  • Physics: Solving equations is essential in physics to describe the motion of objects and the forces acting on them.
  • Engineering: Solving equations is used in engineering to design and optimize systems, such as bridges and buildings.
  • Economics: Solving equations is used in economics to model the behavior of economic systems and make predictions about future trends.

Q: Can you provide an example of a real-world application of solving equations?

A: Yes, here's an example of a real-world application of solving equations:

Suppose you are a civil engineer designing a bridge. You need to determine the length of the bridge that will allow it to span a certain distance. You can use the equation:

L = 2d

where L is the length of the bridge and d is the distance it needs to span.

To solve for L, you can divide both sides of the equation by 2:

L/2 = d

L = 2d

This equation tells you that the length of the bridge is twice the distance it needs to span.

Conclusion

Solving equations is a fundamental concept in mathematics, and it's essential to understand the steps involved to arrive at the correct solution. By following the correct steps and using the correct reciprocal, we can ensure that we arrive at the correct solution to the equation.