Cedric And Insha Solved The Same Equation Using The Calculations Below. \[ \begin{tabular}{|c|c|} \hline \textbf{Cedric's Work} & \textbf{Insha's Work} \\ \hline Z-4.5=-1.5$ & Z − 4.5 = − 1.5 Z-4.5=-1.5 Z − 4.5 = − 1.5 \ Z − 4.5 + 4.5 = − 1.5 + 4.5 Z-4.5+4.5=-1.5+4.5 Z − 4.5 + 4.5 = − 1.5 + 4.5 & Z − 4.5 + ( − 4.5 ) = − 1.5 + ( − 4.5 ) Z-4.5+(-4.5)=-1.5+(-4.5) Z − 4.5 + ( − 4.5 ) = − 1.5 + ( − 4.5 )

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Introduction

In mathematics, solving linear equations is a fundamental concept that forms the basis of various mathematical operations. Two students, Cedric and Insha, were given the same equation to solve, and their calculations are presented below. In this article, we will analyze their work, identify any errors or discrepancies, and provide a step-by-step solution to the equation.

The Equation

The equation given to both Cedric and Insha is:

z−4.5=−1.5z - 4.5 = -1.5

Cedric's Work

Cedric's work is presented below:

Cedric's Work Insha's Work
z−4.5=−1.5z - 4.5 = -1.5 z−4.5=−1.5z - 4.5 = -1.5
z−4.5+4.5=−1.5+4.5z - 4.5 + 4.5 = -1.5 + 4.5 z−4.5+(−4.5)=−1.5+(−4.5)z - 4.5 + (-4.5) = -1.5 + (-4.5)

Analysis of Cedric's Work

Cedric's work appears to be correct, as he has added 4.5 to both sides of the equation to isolate the variable z. However, we need to verify if this is the correct approach and if there are any errors in his calculations.

Step 1: Add 4.5 to both sides of the equation

Cedric's first step is to add 4.5 to both sides of the equation:

z−4.5+4.5=−1.5+4.5z - 4.5 + 4.5 = -1.5 + 4.5

This is a correct step, as adding 4.5 to both sides of the equation will eliminate the -4.5 term and isolate the variable z.

Step 2: Simplify the equation

Cedric's next step is to simplify the equation:

z=3z = 3

This is also a correct step, as the -4.5 and +4.5 terms cancel each other out, leaving only the variable z.

Insha's Work

Insha's work is presented below:

Cedric's Work Insha's Work
z−4.5=−1.5z - 4.5 = -1.5 z−4.5=−1.5z - 4.5 = -1.5
z−4.5+(−4.5)=−1.5+(−4.5)z - 4.5 + (-4.5) = -1.5 + (-4.5) z−4.5+(−4.5)=−1.5+(−4.5)z - 4.5 + (-4.5) = -1.5 + (-4.5)

Analysis of Insha's Work

Insha's work appears to be incorrect, as she has added -4.5 to both sides of the equation instead of adding 4.5. This will result in an incorrect solution.

Step 1: Add -4.5 to both sides of the equation

Insha's first step is to add -4.5 to both sides of the equation:

z−4.5+(−4.5)=−1.5+(−4.5)z - 4.5 + (-4.5) = -1.5 + (-4.5)

This is an incorrect step, as adding -4.5 to both sides of the equation will eliminate the -4.5 term, but it will also change the sign of the -1.5 term.

Step 2: Simplify the equation

Insha's next step is to simplify the equation:

z=−6z = -6

This is also an incorrect step, as the -4.5 and +4.5 terms cancel each other out, but the -1.5 term is changed to -6.

Conclusion

In conclusion, Cedric's work is correct, and he has successfully solved the equation. Insha's work, however, is incorrect, and she has made an error in her calculations. The correct solution to the equation is:

z=3z = 3

Tips for Solving Linear Equations

When solving linear equations, it is essential to follow the correct steps and avoid making errors. Here are some tips to help you solve linear equations correctly:

  • Read the equation carefully: Before starting to solve the equation, read it carefully and understand what is being asked.
  • Identify the variable: Identify the variable that you need to solve for and isolate it on one side of the equation.
  • Add or subtract the same value to both sides: Add or subtract the same value to both sides of the equation to eliminate the constant term.
  • Simplify the equation: Simplify the equation by combining like terms and eliminating any unnecessary terms.
  • Check your solution: Check your solution by plugging it back into the original equation to ensure that it is correct.

By following these tips and being careful with your calculations, you can successfully solve linear equations and become proficient in mathematics.

Common Mistakes to Avoid

When solving linear equations, there are several common mistakes to avoid. Here are some of the most common mistakes:

  • Adding or subtracting the wrong value: Adding or subtracting the wrong value to both sides of the equation can result in an incorrect solution.
  • Not simplifying the equation: Not simplifying the equation can make it difficult to identify the solution and can lead to errors.
  • Not checking the solution: Not checking the solution by plugging it back into the original equation can result in an incorrect solution.

By being aware of these common mistakes and taking steps to avoid them, you can improve your skills in solving linear equations and become more confident in your abilities.

Real-World Applications

Linear equations have numerous real-world applications in various fields, including:

  • Science: Linear equations are used to model real-world phenomena, such as the motion of objects, the growth of populations, and the behavior of electrical circuits.
  • Engineering: Linear equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
  • Economics: Linear equations are used to model economic systems, such as supply and demand, and to make predictions about future economic trends.

By understanding how to solve linear equations, you can apply this knowledge to real-world problems and become proficient in mathematics.

Conclusion

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. It is a simple equation that can be solved using basic algebraic operations.

Q: What are the steps to solve a linear equation?

A: The steps to solve a linear equation are:

  1. Read the equation carefully: Before starting to solve the equation, read it carefully and understand what is being asked.
  2. Identify the variable: Identify the variable that you need to solve for and isolate it on one side of the equation.
  3. Add or subtract the same value to both sides: Add or subtract the same value to both sides of the equation to eliminate the constant term.
  4. Simplify the equation: Simplify the equation by combining like terms and eliminating any unnecessary terms.
  5. Check your solution: Check your solution by plugging it back into the original equation to ensure that it is correct.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1, while a quadratic equation is an equation in which the highest power of the variable(s) is 2. Linear equations can be solved using basic algebraic operations, while quadratic equations require more advanced techniques, such as factoring or the quadratic formula.

Q: How do I know if an equation is linear or quadratic?

A: To determine if an equation is linear or quadratic, look at the highest power of the variable(s). If the highest power is 1, the equation is linear. If the highest power is 2, the equation is quadratic.

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

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

  • Adding or subtracting the wrong value: Adding or subtracting the wrong value to both sides of the equation can result in an incorrect solution.
  • Not simplifying the equation: Not simplifying the equation can make it difficult to identify the solution and can lead to errors.
  • Not checking the solution: Not checking the solution by plugging it back into the original equation can result in an incorrect solution.

Q: How do I check my solution to a linear equation?

A: To check your solution to a linear equation, plug the solution back into the original equation and verify that it is true. If the solution is correct, the equation will be true. If the solution is incorrect, the equation will be false.

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

A: Linear equations have numerous real-world applications in various fields, including:

  • Science: Linear equations are used to model real-world phenomena, such as the motion of objects, the growth of populations, and the behavior of electrical circuits.
  • Engineering: Linear equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
  • Economics: Linear equations are used to model economic systems, such as supply and demand, and to make predictions about future economic trends.

Q: How can I practice solving linear equations?

A: There are many ways to practice solving linear equations, including:

  • Working through practice problems: Practice solving linear equations by working through practice problems in a textbook or online resource.
  • Using online resources: Use online resources, such as Khan Academy or Mathway, to practice solving linear equations.
  • Solving real-world problems: Solve real-world problems that involve linear equations, such as calculating the cost of a product or determining the amount of time it will take to complete a task.

By following these tips and practicing regularly, you can become proficient in solving linear equations and apply this knowledge to real-world problems.