Let's analyze each of the given equations step-by-step to identify the one that has no solution.
### Equation 1
[tex]\[ 13y + 2 - 2y = 10y + 3 - y \][/tex]
1. Combine like terms:
[tex]\[ (13y - 2y) + 2 = (10y - y) + 3 \][/tex]
[tex]\[ 11y + 2 = 9y + 3 \][/tex]
2. Move all terms involving [tex]\(y\)[/tex] to one side and constants to the other:
[tex]\[ 11y - 9y = 3 - 2 \][/tex]
[tex]\[ 2y = 1 \][/tex]
3. Solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{1}{2} \][/tex]
The solution is [tex]\( y = \frac{1}{2} \)[/tex].
### Equation 2
[tex]\[ 9(3y + 7) - 2 = 3(-9y + 9) \][/tex]
1. Distribute the constants:
[tex]\[ 27y + 63 - 2 = -27y + 27 \][/tex]
2. Simplify both sides:
[tex]\[ 27y + 61 = -27y + 27 \][/tex]
3. Move all terms involving [tex]\(y\)[/tex] to one side and constants to the other:
[tex]\[ 27y + 27y = 27 - 61 \][/tex]
[tex]\[ 54y = -34 \][/tex]
4. Solve for [tex]\( y \)[/tex]:
[tex]\[ y = -\frac{34}{54} \][/tex]
[tex]\[ y = -\frac{17}{27} \][/tex]
The solution is [tex]\( y = -\frac{17}{27} \)[/tex].
### Equation 3
[tex]\[ 32.1y + 3.1 + 2.4y - 8.2 = 34.5y - 5.1 \][/tex]
1. Combine like terms:
[tex]\[ (32.1y + 2.4y) + (3.1 - 8.2) = 34.5y - 5.1 \][/tex]
[tex]\[ 34.5y - 5.1 = 34.5y - 5.1 \][/tex]
This equality holds true for all [tex]\( y \)[/tex]. Thus, infinitely many solutions exist for this equation.
### Equation 4
[tex]\[ 5(2.2y + 3.4) = 5(y - 2) + 6y \][/tex]
1. Distribute the constants:
[tex]\[ 11y + 17 = 5y - 10 + 6y \][/tex]
2. Combine like terms:
[tex]\[ 11y + 17 = 11y - 10 \][/tex]
3. Move all terms involving [tex]\(y\)[/tex] to one side and constants to the other:
[tex]\[ 11y - 11y + 17 = -10 \][/tex]
[tex]\[ 17 = -10 \][/tex]
This is a contradiction, meaning that there is no value of [tex]\( y \)[/tex] that can satisfy this equation.
### Conclusion
The equation with no solution is:
[tex]\[ 5(2.2 y + 3.4) = 5(y - 2) + 6 y \][/tex]
Thus, the correct answer from the drop-down menu is Equation 1.
