Clara Solved The Equation 7 3 X = − 2 3 \frac{7}{3} X = -\frac{2}{3} 3 7 ​ X = − 3 2 ​ As Shown: 7 3 × ( 3 7 ) = − 2 3 ( 3 7 ) X = − 14 \begin{aligned} \frac{7}{3} \times \left(\frac{3}{7}\right) & = -\frac{2}{3} \left(\frac{3}{7}\right) \\ x & = -14 \end{aligned} 3 7 ​ × ( 7 3 ​ ) X ​ = − 3 2 ​ ( 7 3 ​ ) = − 14 ​ What Is Clara's Error?A. Clara

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Introduction

Solving equations is a fundamental concept in mathematics, and it requires careful attention to detail and a thorough understanding of the underlying principles. In this article, we will examine the work of Clara, a student who attempted to solve the equation 73x=−23\frac{7}{3} x = -\frac{2}{3}. Unfortunately, Clara made an error in her solution, and we will identify the mistake and provide a correct solution.

The Equation

The equation that Clara attempted to solve is 73x=−23\frac{7}{3} x = -\frac{2}{3}. This is a simple linear equation, and it can be solved by isolating the variable xx.

Clara's Solution

Clara's solution is shown below:

73×(37)=−23(37)x=−14\begin{aligned} \frac{7}{3} \times \left(\frac{3}{7}\right) & = -\frac{2}{3} \left(\frac{3}{7}\right) \\ x & = -14 \end{aligned}

Error Analysis

At first glance, Clara's solution appears to be correct. However, upon closer inspection, we can see that there is a mistake. The error occurs in the first line of the solution, where Clara multiplied both sides of the equation by 37\frac{3}{7}.

The Correct Solution

To solve the equation 73x=−23\frac{7}{3} x = -\frac{2}{3}, we need to isolate the variable xx. We can do this by multiplying both sides of the equation by the reciprocal of 73\frac{7}{3}, which is 37\frac{3}{7}.

73x=−2337×73x=37×−23x=−27\begin{aligned} \frac{7}{3} x & = -\frac{2}{3} \\ \frac{3}{7} \times \frac{7}{3} x & = \frac{3}{7} \times -\frac{2}{3} \\ x & = -\frac{2}{7} \end{aligned}

Conclusion

In conclusion, Clara's error in solving the equation 73x=−23\frac{7}{3} x = -\frac{2}{3} was in multiplying both sides of the equation by 37\frac{3}{7} instead of the reciprocal of 73\frac{7}{3}. The correct solution is x=−27x = -\frac{2}{7}.

Common Mistakes in Solving Equations

There are several common mistakes that students make when solving equations. Some of these mistakes include:

  • Multiplying both sides of the equation by the wrong factor: This is the mistake that Clara made in the example above. When multiplying both sides of the equation by a factor, it is essential to ensure that the factor is the reciprocal of the coefficient of the variable.
  • Dividing both sides of the equation by the wrong factor: This is another common mistake that students make when solving equations. When dividing both sides of the equation by a factor, it is essential to ensure that the factor is the coefficient of the variable.
  • Not checking the solution: It is essential to check the solution to an equation to ensure that it is correct. This can be done by plugging the solution back into the original equation and checking that it is true.

Tips for Solving Equations

Solving equations can be a challenging task, but there are several tips that can help make it easier. Some of these tips include:

  • Read the equation carefully: Before attempting to solve an equation, it is essential to read it carefully and understand what it is asking for.
  • Identify the variable: The variable is the letter or symbol that represents the unknown value. It is essential to identify the variable and isolate it in the equation.
  • Use the correct operations: When solving an equation, it is essential to use the correct operations. For example, when multiplying both sides of the equation by a factor, it is essential to use the multiplication symbol (×\times) instead of the division symbol (÷\div).
  • Check the solution: It is essential to check the solution to an equation to ensure that it is correct. This can be done by plugging the solution back into the original equation and checking that it is true.

Conclusion

Q: What is the first step in solving an equation?

A: The first step in solving an equation is to read the equation carefully and understand what it is asking for. This includes identifying the variable and the constant terms.

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 is 1, whereas a quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I solve a linear equation?

A: To solve a linear equation, you can use the following steps:

  1. Isolate the variable by adding or subtracting the same value to both sides of the equation.
  2. Multiply or divide both sides of the equation by the same value to eliminate the coefficient of the variable.
  3. Check the solution by plugging it back into the original equation.

Q: What is the order of operations when solving an equation?

A: The order of operations when solving an equation is:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I handle fractions when solving an equation?

A: When solving an equation with fractions, you can multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions.

Q: What is the difference between a direct and inverse variation?

A: A direct variation is a relationship between two variables in which one variable is a constant multiple of the other variable. An inverse variation is a relationship between two variables in which one variable is a constant divided by the other variable.

Q: How do I solve an equation with a variable in the denominator?

A: To solve an equation with a variable in the denominator, you can multiply both sides of the equation by the reciprocal of the denominator to eliminate the variable from the denominator.

Q: What is the importance of checking the solution to an equation?

A: Checking the solution to an equation is essential to ensure that it is correct. This can be done by plugging the solution back into the original equation and checking that it is true.

Q: How do I handle equations with absolute values?

A: When solving an equation with absolute values, you need to consider both the positive and negative cases. This means that you need to solve the equation for both the positive and negative values of the absolute value.

Q: What is the difference between a system of linear equations and a system of nonlinear equations?

A: A system of linear equations is a set of two or more linear equations that are solved simultaneously. A system of nonlinear equations is a set of two or more nonlinear equations that are solved simultaneously.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, you can use the following methods:

  1. Substitution method: Substitute the expression for one variable from one equation into the other equation.
  2. Elimination method: Add or subtract the equations to eliminate one variable.
  3. Graphical method: Graph the equations on a coordinate plane and find the point of intersection.

Q: What is the importance of graphing equations?

A: Graphing equations is an essential tool for visualizing the relationship between the variables. It can help you identify the solution to an equation and understand the behavior of the equation.

Q: How do I graph a linear equation?

A: To graph a linear equation, you can use the following steps:

  1. Find the x-intercept by setting y = 0 and solving for x.
  2. Find the y-intercept by setting x = 0 and solving for y.
  3. Plot the points on a coordinate plane and draw a line through them.

Q: What is the difference between a function and a relation?

A: A function is a relation in which each input corresponds to exactly one output. A relation is a set of ordered pairs that satisfy a certain condition.

Q: How do I determine if a relation is a function?

A: To determine if a relation is a function, you can use the following criteria:

  1. Each input corresponds to exactly one output.
  2. No input corresponds to more than one output.
  3. No output corresponds to more than one input.