Answer :
Sure, let's carefully analyze each equation step-by-step to determine which one has infinitely many solutions.
### Equation A:
[tex]\[ -6.8 + 3y + 2.4 = 4.3 - 3y \][/tex]
1. Combine like terms on the left-hand side:
[tex]\[ -6.8 + 2.4 + 3y = -4.4 + 3y \][/tex]
[tex]\[ -4.4 + 3y = 4.3 - 3y \][/tex]
2. Move all \( y \)-terms to one side:
[tex]\[ 3y + 3y = 4.3 + 4.4 \][/tex]
[tex]\[ 6y = 8.7 \][/tex]
3. Simplify:
[tex]\[ y = \frac{8.7}{6} = 1.45 \][/tex]
This equation has a single solution.
### Equation B:
[tex]\[ \frac{1}{3}y + 2.5 - \frac{2}{3}y = 1.2 \][/tex]
1. Combine like terms:
[tex]\[ (\frac{1}{3} - \frac{2}{3})y + 2.5 = 1.2 \][/tex]
[tex]\[ -\frac{1}{3}y + 2.5 = 1.2 \][/tex]
2. Move constant terms to the other side:
[tex]\[ -\frac{1}{3}y = 1.2 - 2.5 \][/tex]
[tex]\[ -\frac{1}{3}y = -1.3 \][/tex]
3. Simplify:
[tex]\[ y = \frac{-1.3 \cdot 3}{-1} \][/tex]
[tex]\[ y = 3.9 \][/tex]
This equation has a single solution.
### Equation C:
[tex]\[ 5.1 + 2y + 1.2 = -2 + 2y + 8.3 \][/tex]
1. Combine like terms on both sides:
[tex]\[ 5.1 + 1.2 + 2y = 6.1 + 2y \][/tex]
[tex]\[ 6.3 + 2y = 6.3 + 2y \][/tex]
We observe here that both sides are identical:
[tex]\[ 6.3 + 2y = 6.3 + 2y \][/tex]
Since this equation holds true for any value of \( y \), it has infinitely many solutions.
### Equation D:
[tex]\[ \frac{2}{5} y = 2.3 + \frac{3}{2} y \][/tex]
1. Move all \( y \)-terms to one side:
[tex]\[ \frac{2}{5} y - \frac{3}{2} y = 2.3 \][/tex]
[tex]\[ \left( \frac{4}{10} - \frac{15}{10} \right) y = 2.3 \][/tex]
[tex]\[ -\frac{11}{10} y = 2.3 \][/tex]
2. Simplify:
[tex]\[ y = \frac{-2.3 \cdot 10}{11} \][/tex]
[tex]\[ y = -\frac{23}{11} = -2.09 \][/tex]
This equation has a single solution.
### Conclusion
From the detailed analysis, we see that Equation C has infinitely many solutions. Therefore, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
### Equation A:
[tex]\[ -6.8 + 3y + 2.4 = 4.3 - 3y \][/tex]
1. Combine like terms on the left-hand side:
[tex]\[ -6.8 + 2.4 + 3y = -4.4 + 3y \][/tex]
[tex]\[ -4.4 + 3y = 4.3 - 3y \][/tex]
2. Move all \( y \)-terms to one side:
[tex]\[ 3y + 3y = 4.3 + 4.4 \][/tex]
[tex]\[ 6y = 8.7 \][/tex]
3. Simplify:
[tex]\[ y = \frac{8.7}{6} = 1.45 \][/tex]
This equation has a single solution.
### Equation B:
[tex]\[ \frac{1}{3}y + 2.5 - \frac{2}{3}y = 1.2 \][/tex]
1. Combine like terms:
[tex]\[ (\frac{1}{3} - \frac{2}{3})y + 2.5 = 1.2 \][/tex]
[tex]\[ -\frac{1}{3}y + 2.5 = 1.2 \][/tex]
2. Move constant terms to the other side:
[tex]\[ -\frac{1}{3}y = 1.2 - 2.5 \][/tex]
[tex]\[ -\frac{1}{3}y = -1.3 \][/tex]
3. Simplify:
[tex]\[ y = \frac{-1.3 \cdot 3}{-1} \][/tex]
[tex]\[ y = 3.9 \][/tex]
This equation has a single solution.
### Equation C:
[tex]\[ 5.1 + 2y + 1.2 = -2 + 2y + 8.3 \][/tex]
1. Combine like terms on both sides:
[tex]\[ 5.1 + 1.2 + 2y = 6.1 + 2y \][/tex]
[tex]\[ 6.3 + 2y = 6.3 + 2y \][/tex]
We observe here that both sides are identical:
[tex]\[ 6.3 + 2y = 6.3 + 2y \][/tex]
Since this equation holds true for any value of \( y \), it has infinitely many solutions.
### Equation D:
[tex]\[ \frac{2}{5} y = 2.3 + \frac{3}{2} y \][/tex]
1. Move all \( y \)-terms to one side:
[tex]\[ \frac{2}{5} y - \frac{3}{2} y = 2.3 \][/tex]
[tex]\[ \left( \frac{4}{10} - \frac{15}{10} \right) y = 2.3 \][/tex]
[tex]\[ -\frac{11}{10} y = 2.3 \][/tex]
2. Simplify:
[tex]\[ y = \frac{-2.3 \cdot 10}{11} \][/tex]
[tex]\[ y = -\frac{23}{11} = -2.09 \][/tex]
This equation has a single solution.
### Conclusion
From the detailed analysis, we see that Equation C has infinitely many solutions. Therefore, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
