Let's solve each equation step-by-step to determine which has no solution.
### Equation A
[tex]\[ 2.3 y + 2 + 3.1 y = 4.3 y + 1.6 + 1.1 y + 0.4 \][/tex]
Combine like terms on both sides of the equation:
[tex]\[ (2.3 y + 3.1 y) + 2 = (4.3 y + 1.1 y) + (1.6 + 0.4) \][/tex]
[tex]\[ 5.4 y + 2 = 5.4 y + 2 \][/tex]
Subtract [tex]\(5.4 y\)[/tex] from both sides:
[tex]\[ 2 = 2 \][/tex]
This equation is always true for any value of [tex]\(y\)[/tex]. Therefore, it has infinitely many solutions.
### Equation B
[tex]\[ 32 x + 25 - 21 x = 10 x \][/tex]
Combine like terms:
[tex]\[ (32 x - 21 x) + 25 = 10 x \][/tex]
[tex]\[ 11 x + 25 = 10 x \][/tex]
Subtract [tex]\(10 x\)[/tex] from both sides:
[tex]\[ 11 x - 10 x + 25 = 0 \][/tex]
[tex]\[ x + 25 = 0 \][/tex]
Subtract 25 from both sides:
[tex]\[ x = -25 \][/tex]
This equation has a unique solution [tex]\( x = -25 \)[/tex].
### Equation C
[tex]\[ \frac{5}{8} x + 2.5 = \frac{3}{8} x + 1.5 + \frac{1}{4} x \][/tex]
Express [tex]\(\frac{1}{4} x\)[/tex] in terms of eighths:
[tex]\[ \frac{1}{4} x = \frac{2}{8} x \][/tex]
Rewrite the equation:
[tex]\[ \frac{5}{8} x + 2.5 = \frac{3}{8} x + 1.5 + \frac{2}{8} x \][/tex]
[tex]\[ \frac{5}{8} x + 2.5 = \left( \frac{3}{8} x + \frac{2}{8} x \right) + 1.5 \][/tex]
[tex]\[ \frac{5}{8} x + 2.5 = \frac{5}{8} x + 1.5 \][/tex]
Subtract [tex]\(\frac{5}{8} x\)[/tex] from both sides:
[tex]\[ 2.5 = 1.5 \][/tex]
This is a contradiction, which means there is no solution.
### Equation D
[tex]\[ \frac{1}{3} + \frac{1}{7} y = \frac{3}{7} y \][/tex]
Subtract [tex]\(\frac{1}{7} y\)[/tex] from both sides:
[tex]\[ \frac{1}{3} = \frac{3}{7} y - \frac{1}{7} y \][/tex]
[tex]\[ \frac{1}{3} = \frac{2}{7} y \][/tex]
Multiply both sides by [tex]\(\frac{7}{2}\)[/tex]:
[tex]\[ y = \frac{7}{2} \cdot \frac{1}{3} \][/tex]
[tex]\[ y = \frac{7}{6} \][/tex]
This equation has a unique solution [tex]\( y = \frac{7}{6} \)[/tex].
### Conclusion
The correct answer is:
C. [tex]\(\frac{5}{8} x + 2.5 = \frac{3}{8} x + 1.5 + \frac{1}{4} x\)[/tex]
