Certainly! Let's analyze each equation to determine which ones have infinitely many solutions by simplifying them and checking their consistency.
### Equation A:
[tex]\[ -10x - 10 = -10x + 10 \][/tex]
1. Start by moving all the terms involving [tex]\( x \)[/tex] to one side and constant terms to the other side.
[tex]\[ -10x - 10 + 10x = -10x + 10 + 10x \][/tex]
2. This simplifies to:
[tex]\[ -10 = 10 \][/tex]
This is a contradiction, as [tex]\(-10\)[/tex] cannot equal [tex]\(10\)[/tex]. Therefore, Equation A does not have infinitely many solutions.
### Equation B:
[tex]\[ 10x - 10 = -10x - 10 \][/tex]
1. Combine like terms by moving [tex]\( x \)[/tex]-terms to one side:
[tex]\[ 10x + 10x - 10 = -10 \][/tex]
2. Combine constants:
[tex]\[ 20x - 10 = -10 \][/tex]
3. Add 10 to both sides:
[tex]\[ 20x = 0 \][/tex]
4. Divide by 20:
[tex]\[ x = 0 \][/tex]
This equation has exactly one solution, [tex]\( x = 0 \)[/tex]. Therefore, Equation B does not have infinitely many solutions.
### Equation C:
[tex]\[ -10x - 10 = -10x - 10 \][/tex]
1. Simplify both sides to see if the equation holds:
[tex]\[ -10x - 10 + 10x = -10x - 10 + 10x \][/tex]
2. This results in:
[tex]\[ -10 = -10 \][/tex]
This is always true, regardless of the value of [tex]\( x \)[/tex]. So, this equation is an identity, meaning it is valid for all values of [tex]\( x \)[/tex]. Therefore, Equation C has infinitely many solutions.
### Equation D:
[tex]\[ 10x - 10 = -10x + 10 \][/tex]
1. Move [tex]\( x \)[/tex] terms to one side:
[tex]\[ 10x + 10x - 10 = 10 \][/tex]
2. Combine constants:
[tex]\[ 20x - 10 = 10 \][/tex]
3. Add 10 to both sides:
[tex]\[ 20x = 20 \][/tex]
4. Divide by 20:
[tex]\[ x = 1 \][/tex]
This equation has exactly one solution, [tex]\( x = 1 \)[/tex]. Therefore, Equation D does not have infinitely many solutions.
### Conclusion
The only equation among the given options that has infinitely many solutions is:
[tex]\[ \text{(C)} -10x - 10 = -10x - 10 \][/tex]
Thus, the answer to the question is:
[tex]\[
\text{(C)}
\][/tex]
