To determine which of the given equations has no solution, let's solve each equation step-by-step.
### Equation A: [tex]\(-4(y+7)=2(-2y-9)-10\)[/tex]
1. Distribute the expressions:
[tex]\[
-4(y + 7) = -4y - 28
\][/tex]
[tex]\[
2(-2y - 9) = -4y - 18
\][/tex]
2. Simplify the equation:
[tex]\[
-4(y + 7) = -4y - 28 \quad \text{and} \quad 2(-2y - 9) - 10 = -4y - 18 - 10 = -4y - 28
\][/tex]
[tex]\[
-4y - 28 = -4y - 28
\][/tex]
This equation is an identity, so it is satisfied for all values of [tex]\(y\)[/tex]. Therefore, Equation A has infinite solutions.
### Equation B: [tex]\(-2(2y+8)=4y+5+y\)[/tex]
1. Distribute the expressions:
[tex]\[
-2(2y + 8) = -4y - 16
\][/tex]
[tex]\[
4y + 5 + y = 5y + 5
\][/tex]
2. Simplify the equation:
[tex]\[
-4y - 16 = 5y + 5
\][/tex]
3. Combine like terms:
[tex]\[
-4y - 5y = 5 + 16
\][/tex]
[tex]\[
-9y = 21
\][/tex]
[tex]\[
y = -\frac{21}{9} = -\frac{7}{3}
\][/tex]
This equation has a unique solution: [tex]\(y = -\frac{7}{3}\)[/tex].
### Equation C: [tex]\(4(y+9)=-4(y-9)\)[/tex]
1. Distribute the expressions:
[tex]\[
4(y + 9) = 4y + 36
\][/tex]
[tex]\[
-4(y - 9) = -4y + 36
\][/tex]
2. Simplify the equation:
[tex]\[
4y + 36 = -4y + 36
\][/tex]
3. Combine like terms:
[tex]\[
4y + 4y = 36 - 36
\][/tex]
[tex]\[
8y = 0
\][/tex]
[tex]\[
y = 0
\][/tex]
This equation has a unique solution: [tex]\(y = 0\)[/tex].
### Equation D: [tex]\(3y+5-7y=4(-y+1)+5\)[/tex]
1. Distribute the expressions:
[tex]\[
3y + 5 - 7y = -4y + 4 + 5
\][/tex]
2. Simplify the equation:
[tex]\[
3y - 7y + 5 = -4y + 9
\][/tex]
[tex]\[
-4y + 5 = -4y + 9
\][/tex]
3. Combine like terms:
[tex]\[
5 = 9
\][/tex]
Since the equation [tex]\(5 = 9\)[/tex] is a contradiction (it is never true), Equation D has no solution.
### Conclusion
Equation D: [tex]\(3y + 5 - 7y = 4(-y + 1) + 5\)[/tex] has no solution. Therefore, the correct answer is:
D. [tex]\(\boxed{3y+5-7y=4(-y+1)+5}\)[/tex]
