Answer :
To determine Clara's error in solving the equation [tex]\(\frac{7}{3} x = -\frac{2}{3}\)[/tex], let's go through the steps to solve it correctly:
1. Original Equation: [tex]\(\frac{7}{3} x = -\frac{2}{3}\)[/tex].
2. Isolate [tex]\(x\)[/tex]: To solve for [tex]\(x\)[/tex], we need to get rid of the fraction [tex]\(\frac{7}{3}\)[/tex] that's multiplying [tex]\(x\)[/tex].
3. Multiply by the Reciprocal: Multiply both sides of the equation by the reciprocal of [tex]\(\frac{7}{3}\)[/tex], which is [tex]\(\frac{3}{7}\)[/tex].
[tex]\[
\left(\frac{7}{3} x\right) \times \left(\frac{3}{7}\right) = \left(-\frac{2}{3}\right) \times \left(\frac{3}{7}\right)
\][/tex]
4. Simplifying:
- On the left side: [tex]\(\frac{7}{3} \times \frac{3}{7} = 1\)[/tex], so we get [tex]\(x\)[/tex].
- On the right side: Calculate [tex]\(-\frac{2}{3} \times \frac{3}{7} = -\frac{6}{21}\)[/tex].
5. Simplified Result: Simplify [tex]\(-\frac{6}{21}\)[/tex]. Both 6 and 21 are divisible by 3.
[tex]\[
-\frac{6}{21} = -\frac{2}{7}
\][/tex]
6. Conclusion: So the correct solution is [tex]\(x = -\frac{2}{7}\)[/tex].
Clara's Error: Clara's error arose from an incorrect simplification after multiplying by the reciprocal. She concluded that the solution was [tex]\(x = -14\)[/tex], which is incorrect. The correct solution is [tex]\(x = -\frac{2}{7}\)[/tex].
1. Original Equation: [tex]\(\frac{7}{3} x = -\frac{2}{3}\)[/tex].
2. Isolate [tex]\(x\)[/tex]: To solve for [tex]\(x\)[/tex], we need to get rid of the fraction [tex]\(\frac{7}{3}\)[/tex] that's multiplying [tex]\(x\)[/tex].
3. Multiply by the Reciprocal: Multiply both sides of the equation by the reciprocal of [tex]\(\frac{7}{3}\)[/tex], which is [tex]\(\frac{3}{7}\)[/tex].
[tex]\[
\left(\frac{7}{3} x\right) \times \left(\frac{3}{7}\right) = \left(-\frac{2}{3}\right) \times \left(\frac{3}{7}\right)
\][/tex]
4. Simplifying:
- On the left side: [tex]\(\frac{7}{3} \times \frac{3}{7} = 1\)[/tex], so we get [tex]\(x\)[/tex].
- On the right side: Calculate [tex]\(-\frac{2}{3} \times \frac{3}{7} = -\frac{6}{21}\)[/tex].
5. Simplified Result: Simplify [tex]\(-\frac{6}{21}\)[/tex]. Both 6 and 21 are divisible by 3.
[tex]\[
-\frac{6}{21} = -\frac{2}{7}
\][/tex]
6. Conclusion: So the correct solution is [tex]\(x = -\frac{2}{7}\)[/tex].
Clara's Error: Clara's error arose from an incorrect simplification after multiplying by the reciprocal. She concluded that the solution was [tex]\(x = -14\)[/tex], which is incorrect. The correct solution is [tex]\(x = -\frac{2}{7}\)[/tex].
