Answer :
- Clara correctly multiplied both sides of the equation by $\frac{3}{7}$ to isolate $x$.
- The correct solution to the equation is $x = -\frac{2}{7}$.
- The error is that the solution is $-\frac{2}{7}$, not -14.
- Therefore, Clara's error lies in the final interpretation of the solution. $\boxed{-\frac{2}{7}}$
### Explanation
1. Analyzing the Problem
The problem is to identify the error Clara made while solving the equation $\frac{7}{3}x = -\frac{2}{3}$. Clara multiplied both sides of the equation by $\frac{3}{7}$. We need to determine if this was the correct operation and if she executed it correctly.
2. Identifying the Correct Operation
To isolate $x$ in the equation $\frac{7}{3}x = -\frac{2}{3}$, we need to perform an operation that will result in $x$ being by itself on one side of the equation. Since $x$ is multiplied by $\frac{7}{3}$, we can multiply both sides of the equation by the reciprocal of $\frac{7}{3}$, which is $\frac{3}{7}$.
3. Evaluating the Result
Clara multiplied both sides by $\frac{3}{7}$, so the operation she performed was correct. Let's check the result of her operation:\n$$\frac{7}{3}x \cdot \frac{3}{7} = x$$ and $$\-\frac{2}{3} \cdot \frac{3}{7} = -\frac{2 \cdot 3}{3 \cdot 7} = -\frac{6}{21} = -\frac{2}{7}$$\nSo, $x = -\frac{2}{7}$.
4. Identifying Clara's Error
Since Clara multiplied by $\frac{3}{7}$ correctly and arrived at the solution $x = -\frac{2}{7}$, her error is not in the operation itself, but potentially in a misunderstanding of why this operation works or in comparing her solution to the given options. The question asks for Clara's error, and the options are:\n- Clara should have divided by $\frac{3}{7}$.\n- Clara should have added $\frac{7}{3}$.\n- Clara should have multiplied by $\frac{7}{3}$.\n- The solution is $-\frac{2}{7}$, not -14.\nClara correctly multiplied by $\frac{3}{7}$, which is the same as dividing by $\frac{7}{3}$. Adding $\frac{7}{3}$ is incorrect. Multiplying by $\frac{7}{3}$ is also incorrect. The solution $-\frac{2}{7}$ is correct. Therefore, the error is not in the calculation, but in the interpretation of the solution or the operation needed.
5. Final Answer
The correct answer is that the solution is $-\frac{2}{7}$, not -14. Clara performed the correct operation, but the question is designed to check if she understands the result.
### Examples
When solving for a variable in an equation, you need to isolate the variable by performing inverse operations. This is similar to balancing a scale; whatever you do to one side, you must do to the other to maintain the balance. For example, if you're calculating the amount of flour needed for a recipe and you know the ratio of flour to other ingredients, you use similar algebraic steps to find the exact quantity of flour.
- The correct solution to the equation is $x = -\frac{2}{7}$.
- The error is that the solution is $-\frac{2}{7}$, not -14.
- Therefore, Clara's error lies in the final interpretation of the solution. $\boxed{-\frac{2}{7}}$
### Explanation
1. Analyzing the Problem
The problem is to identify the error Clara made while solving the equation $\frac{7}{3}x = -\frac{2}{3}$. Clara multiplied both sides of the equation by $\frac{3}{7}$. We need to determine if this was the correct operation and if she executed it correctly.
2. Identifying the Correct Operation
To isolate $x$ in the equation $\frac{7}{3}x = -\frac{2}{3}$, we need to perform an operation that will result in $x$ being by itself on one side of the equation. Since $x$ is multiplied by $\frac{7}{3}$, we can multiply both sides of the equation by the reciprocal of $\frac{7}{3}$, which is $\frac{3}{7}$.
3. Evaluating the Result
Clara multiplied both sides by $\frac{3}{7}$, so the operation she performed was correct. Let's check the result of her operation:\n$$\frac{7}{3}x \cdot \frac{3}{7} = x$$ and $$\-\frac{2}{3} \cdot \frac{3}{7} = -\frac{2 \cdot 3}{3 \cdot 7} = -\frac{6}{21} = -\frac{2}{7}$$\nSo, $x = -\frac{2}{7}$.
4. Identifying Clara's Error
Since Clara multiplied by $\frac{3}{7}$ correctly and arrived at the solution $x = -\frac{2}{7}$, her error is not in the operation itself, but potentially in a misunderstanding of why this operation works or in comparing her solution to the given options. The question asks for Clara's error, and the options are:\n- Clara should have divided by $\frac{3}{7}$.\n- Clara should have added $\frac{7}{3}$.\n- Clara should have multiplied by $\frac{7}{3}$.\n- The solution is $-\frac{2}{7}$, not -14.\nClara correctly multiplied by $\frac{3}{7}$, which is the same as dividing by $\frac{7}{3}$. Adding $\frac{7}{3}$ is incorrect. Multiplying by $\frac{7}{3}$ is also incorrect. The solution $-\frac{2}{7}$ is correct. Therefore, the error is not in the calculation, but in the interpretation of the solution or the operation needed.
5. Final Answer
The correct answer is that the solution is $-\frac{2}{7}$, not -14. Clara performed the correct operation, but the question is designed to check if she understands the result.
### Examples
When solving for a variable in an equation, you need to isolate the variable by performing inverse operations. This is similar to balancing a scale; whatever you do to one side, you must do to the other to maintain the balance. For example, if you're calculating the amount of flour needed for a recipe and you know the ratio of flour to other ingredients, you use similar algebraic steps to find the exact quantity of flour.
