Answer :
First, we need to solve for [tex]\( y \)[/tex] from the equation [tex]\( 3y = 36 - 5x \)[/tex] and analyze the given work for correctness.
### Step-by-Step Solution:
1. Original Equation:
[tex]\[
3y = 36 - 5x
\][/tex]
2. Divide both sides by 3 to isolate [tex]\( y \)[/tex]:
[tex]\[
\frac{3y}{3} = \frac{36 - 5x}{3}
\][/tex]
3. Simplify both sides:
- For the left-hand side (LHS):
[tex]\[
\frac{3y}{3} = y
\][/tex]
This simplifies to:
[tex]\[
y = \frac{36}{3} - \frac{5x}{3}
\][/tex]
- For the right-hand side (RHS):
[tex]\[
\frac{36}{3} - \frac{5x}{3}
\][/tex]
Simplifying each term individually:
[tex]\[
\frac{36}{3} = 12
\][/tex]
and
[tex]\[
\frac{5x}{3} = \frac{5}{3}x
\][/tex]
4. Combine the simplified terms on the right-hand side:
[tex]\[
y = 12 - \frac{5}{3} x
\][/tex]
So, the equivalent equation for [tex]\( y \)[/tex] is:
[tex]\[
y = 12 - \frac{5}{3} x
\][/tex]
### Analysis of the Given Work:
1. First Choice: "The work was completed correctly."
- This statement is correct. The provided solution shows the correct steps to isolate [tex]\( y \)[/tex] and simplifies both sides properly.
2. Second Choice: "Both sides needed to be multiplied by 3, rather than divided by 3."
- This statement is incorrect. Multiplying both sides by 3 would not isolate [tex]\( y \)[/tex]. We needed to divide both sides by 3 to solve for [tex]\( y \)[/tex].
3. Third Choice: "When dividing 36 by 3, the answer should have been [tex]\( \frac{1}{12} \)[/tex], not 12."
- This statement is incorrect. The correct division of 36 by 3 is indeed 12, not [tex]\( \frac{1}{12} \)[/tex].
4. Fourth Choice: "Both terms on the right side need to be divided by 3, not just the 36."
- This statement is also correct. Both 36 and [tex]\( -5x \)[/tex] were accurately divided by 3 in the original work.
### Conclusion:
- Correct Work Statement: The work was completed correctly.
- Incorrect Work Statements: Both sides needed to be multiplied by 3, rather than divided by 3. When dividing 36 by 3, the answer should have been [tex]\( \frac{1}{12} \)[/tex], not 12. Both terms on the right side need to be divided by 3, not just the 36.
From this analysis, we conclude that the work was done correctly. The correct choice is the first one.
### Step-by-Step Solution:
1. Original Equation:
[tex]\[
3y = 36 - 5x
\][/tex]
2. Divide both sides by 3 to isolate [tex]\( y \)[/tex]:
[tex]\[
\frac{3y}{3} = \frac{36 - 5x}{3}
\][/tex]
3. Simplify both sides:
- For the left-hand side (LHS):
[tex]\[
\frac{3y}{3} = y
\][/tex]
This simplifies to:
[tex]\[
y = \frac{36}{3} - \frac{5x}{3}
\][/tex]
- For the right-hand side (RHS):
[tex]\[
\frac{36}{3} - \frac{5x}{3}
\][/tex]
Simplifying each term individually:
[tex]\[
\frac{36}{3} = 12
\][/tex]
and
[tex]\[
\frac{5x}{3} = \frac{5}{3}x
\][/tex]
4. Combine the simplified terms on the right-hand side:
[tex]\[
y = 12 - \frac{5}{3} x
\][/tex]
So, the equivalent equation for [tex]\( y \)[/tex] is:
[tex]\[
y = 12 - \frac{5}{3} x
\][/tex]
### Analysis of the Given Work:
1. First Choice: "The work was completed correctly."
- This statement is correct. The provided solution shows the correct steps to isolate [tex]\( y \)[/tex] and simplifies both sides properly.
2. Second Choice: "Both sides needed to be multiplied by 3, rather than divided by 3."
- This statement is incorrect. Multiplying both sides by 3 would not isolate [tex]\( y \)[/tex]. We needed to divide both sides by 3 to solve for [tex]\( y \)[/tex].
3. Third Choice: "When dividing 36 by 3, the answer should have been [tex]\( \frac{1}{12} \)[/tex], not 12."
- This statement is incorrect. The correct division of 36 by 3 is indeed 12, not [tex]\( \frac{1}{12} \)[/tex].
4. Fourth Choice: "Both terms on the right side need to be divided by 3, not just the 36."
- This statement is also correct. Both 36 and [tex]\( -5x \)[/tex] were accurately divided by 3 in the original work.
### Conclusion:
- Correct Work Statement: The work was completed correctly.
- Incorrect Work Statements: Both sides needed to be multiplied by 3, rather than divided by 3. When dividing 36 by 3, the answer should have been [tex]\( \frac{1}{12} \)[/tex], not 12. Both terms on the right side need to be divided by 3, not just the 36.
From this analysis, we conclude that the work was done correctly. The correct choice is the first one.
