Answer :
To solve the equation [tex]\(\frac{7}{3} x = -\frac{2}{3}\)[/tex] correctly, let's go through the process step-by-step:
1. Identify the operation needed to solve for [tex]\(x\)[/tex]:
The equation given is [tex]\(\frac{7}{3} x = -\frac{2}{3}\)[/tex]. To isolate [tex]\(x\)[/tex], we need to get rid of the fraction [tex]\(\frac{7}{3}\)[/tex] that is multiplying [tex]\(x\)[/tex].
2. Multiply by the reciprocal:
The simplest way to eliminate the fraction is to multiply both sides of the equation by the reciprocal of [tex]\(\frac{7}{3}\)[/tex], which is [tex]\(\frac{3}{7}\)[/tex].
[tex]\[
\left(\frac{7}{3} \times \frac{3}{7}\right) x = -\frac{2}{3} \times \frac{3}{7}
\][/tex]
3. Simplify the left side of the equation:
[tex]\(\frac{7}{3} \times \frac{3}{7} = 1\)[/tex], so the left side simplifies to:
[tex]\[
1 \times x = x
\][/tex]
4. Simplify the right side of the equation:
Multiply [tex]\(-\frac{2}{3}\)[/tex] by [tex]\(\frac{3}{7}\)[/tex]:
[tex]\[
-\frac{2}{3} \times \frac{3}{7} = -\frac{2 \times 3}{3 \times 7} = -\frac{6}{21}
\][/tex]
Simplifying [tex]\(-\frac{6}{21}\)[/tex], we divide both the numerator and the denominator by their greatest common divisor, which is 3:
[tex]\[
-\frac{6}{21} = -\frac{2}{7}
\][/tex]
5. Write the final solution:
Therefore, the correct solution is:
[tex]\[
x = -\frac{2}{7}
\][/tex]
Clara's mistake was in the final calculation of the solution. She incorrectly stated that [tex]\(x = -14\)[/tex] instead of the correct solution, which is [tex]\(x = -\frac{2}{7}\)[/tex].
1. Identify the operation needed to solve for [tex]\(x\)[/tex]:
The equation given is [tex]\(\frac{7}{3} x = -\frac{2}{3}\)[/tex]. To isolate [tex]\(x\)[/tex], we need to get rid of the fraction [tex]\(\frac{7}{3}\)[/tex] that is multiplying [tex]\(x\)[/tex].
2. Multiply by the reciprocal:
The simplest way to eliminate the fraction is to multiply both sides of the equation by the reciprocal of [tex]\(\frac{7}{3}\)[/tex], which is [tex]\(\frac{3}{7}\)[/tex].
[tex]\[
\left(\frac{7}{3} \times \frac{3}{7}\right) x = -\frac{2}{3} \times \frac{3}{7}
\][/tex]
3. Simplify the left side of the equation:
[tex]\(\frac{7}{3} \times \frac{3}{7} = 1\)[/tex], so the left side simplifies to:
[tex]\[
1 \times x = x
\][/tex]
4. Simplify the right side of the equation:
Multiply [tex]\(-\frac{2}{3}\)[/tex] by [tex]\(\frac{3}{7}\)[/tex]:
[tex]\[
-\frac{2}{3} \times \frac{3}{7} = -\frac{2 \times 3}{3 \times 7} = -\frac{6}{21}
\][/tex]
Simplifying [tex]\(-\frac{6}{21}\)[/tex], we divide both the numerator and the denominator by their greatest common divisor, which is 3:
[tex]\[
-\frac{6}{21} = -\frac{2}{7}
\][/tex]
5. Write the final solution:
Therefore, the correct solution is:
[tex]\[
x = -\frac{2}{7}
\][/tex]
Clara's mistake was in the final calculation of the solution. She incorrectly stated that [tex]\(x = -14\)[/tex] instead of the correct solution, which is [tex]\(x = -\frac{2}{7}\)[/tex].
