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Answer :
To determine which equations have infinitely many solutions, we need to check if simplifying them results in a true statement such as 0 = 0. Let's check each equation step by step:
### Equation A: [tex]\(75x + 57 = -75x + 57\)[/tex]
1. Combine like terms:
- Move [tex]\(75x\)[/tex] from the left side to the right side by adding [tex]\(75x\)[/tex] to both sides:
[tex]\[
75x + 75x + 57 = 57
\][/tex]
- Simplify:
[tex]\[
150x + 57 = 57
\][/tex]
- Subtract 57 from both sides:
[tex]\[
150x = 0
\][/tex]
2. This equation means [tex]\(x = 0\)[/tex] as the solution, so it does not have infinitely many solutions.
### Equation B: [tex]\(-57x + 57 = -75x + 75\)[/tex]
1. Combine like terms:
- Move [tex]\(-57x\)[/tex] from the left side to the right side by adding it to both sides:
[tex]\[
57 = -75x + 57x + 75
\][/tex]
- Simplify the equations:
[tex]\[
57 = -18x + 75
\][/tex]
- Subtract 75 from both sides:
[tex]\[
-18 = -18x
\][/tex]
2. Simplifying further:
- Divide each side by -18:
[tex]\[
x = 1
\][/tex]
3. This equation means [tex]\(x = 1\)[/tex] is a specific solution, so it does not have infinitely many solutions.
### Equation C: [tex]\(-75x + 57 = -75x + 57\)[/tex]
1. This equation is already simplified and both sides are identical, meaning every value of [tex]\(x\)[/tex] satisfies this equation.
2. Therefore, this equation does indeed have infinitely many solutions because it simplifies to a statement like 0 = 0.
Based on the steps, only Equation C has infinitely many solutions.
### Equation A: [tex]\(75x + 57 = -75x + 57\)[/tex]
1. Combine like terms:
- Move [tex]\(75x\)[/tex] from the left side to the right side by adding [tex]\(75x\)[/tex] to both sides:
[tex]\[
75x + 75x + 57 = 57
\][/tex]
- Simplify:
[tex]\[
150x + 57 = 57
\][/tex]
- Subtract 57 from both sides:
[tex]\[
150x = 0
\][/tex]
2. This equation means [tex]\(x = 0\)[/tex] as the solution, so it does not have infinitely many solutions.
### Equation B: [tex]\(-57x + 57 = -75x + 75\)[/tex]
1. Combine like terms:
- Move [tex]\(-57x\)[/tex] from the left side to the right side by adding it to both sides:
[tex]\[
57 = -75x + 57x + 75
\][/tex]
- Simplify the equations:
[tex]\[
57 = -18x + 75
\][/tex]
- Subtract 75 from both sides:
[tex]\[
-18 = -18x
\][/tex]
2. Simplifying further:
- Divide each side by -18:
[tex]\[
x = 1
\][/tex]
3. This equation means [tex]\(x = 1\)[/tex] is a specific solution, so it does not have infinitely many solutions.
### Equation C: [tex]\(-75x + 57 = -75x + 57\)[/tex]
1. This equation is already simplified and both sides are identical, meaning every value of [tex]\(x\)[/tex] satisfies this equation.
2. Therefore, this equation does indeed have infinitely many solutions because it simplifies to a statement like 0 = 0.
Based on the steps, only Equation C has infinitely many solutions.
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