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Answer :
To determine which of these equations have infinitely many solutions, let's analyze each one step by step. An equation will have infinitely many solutions if both sides simplify to the same expression.
Equation A: [tex]\(-57x + 57 = -75x + 75\)[/tex]
1. Add [tex]\(75x\)[/tex] to both sides to eliminate [tex]\(x\)[/tex] on the right:
[tex]\[-57x + 75x + 57 = 75\][/tex]
2. Simplify the left side:
[tex]\[18x + 57 = 75\][/tex]
3. Subtract 57 from both sides:
[tex]\[18x = 18\][/tex]
4. Divide by 18:
[tex]\[x = 1\][/tex]
This equation has a single solution, [tex]\(x = 1\)[/tex].
---
Equation B: [tex]\(75x + 57 = -75x + 57\)[/tex]
1. Add [tex]\(75x\)[/tex] to both sides to eliminate [tex]\(x\)[/tex] on the right:
[tex]\[75x + 75x + 57 = 57\][/tex]
2. Simplify the left side:
[tex]\[150x + 57 = 57\][/tex]
3. Subtract 57 from both sides:
[tex]\[150x = 0\][/tex]
4. Divide by 150:
[tex]\[x = 0\][/tex]
This equation also has a single solution, [tex]\(x = 0\)[/tex].
---
Equation C: [tex]\(-75x + 57 = -75x + 57\)[/tex]
1. Simplify both sides:
[tex]\[-75x + 57 = -75x + 57\][/tex]
Since both sides are identical already, the equation is true for all values of [tex]\(x\)[/tex].
This equation has infinitely many solutions.
---
Equation D: [tex]\(57x + 57 = -75x - 75\)[/tex]
1. Add [tex]\(75x\)[/tex] to both sides to eliminate [tex]\(x\)[/tex] on the right:
[tex]\[57x + 75x + 57 = -75\][/tex]
2. Simplify the left side:
[tex]\[132x + 57 = -75\][/tex]
3. Subtract 57 from both sides:
[tex]\[132x = -132\][/tex]
4. Divide by 132:
[tex]\[x = -1\][/tex]
This equation has a single solution, [tex]\(x = -1\)[/tex].
---
Therefore, the equation with infinitely many solutions is (C) [tex]\(-75x + 57 = -75x + 57\)[/tex].
Equation A: [tex]\(-57x + 57 = -75x + 75\)[/tex]
1. Add [tex]\(75x\)[/tex] to both sides to eliminate [tex]\(x\)[/tex] on the right:
[tex]\[-57x + 75x + 57 = 75\][/tex]
2. Simplify the left side:
[tex]\[18x + 57 = 75\][/tex]
3. Subtract 57 from both sides:
[tex]\[18x = 18\][/tex]
4. Divide by 18:
[tex]\[x = 1\][/tex]
This equation has a single solution, [tex]\(x = 1\)[/tex].
---
Equation B: [tex]\(75x + 57 = -75x + 57\)[/tex]
1. Add [tex]\(75x\)[/tex] to both sides to eliminate [tex]\(x\)[/tex] on the right:
[tex]\[75x + 75x + 57 = 57\][/tex]
2. Simplify the left side:
[tex]\[150x + 57 = 57\][/tex]
3. Subtract 57 from both sides:
[tex]\[150x = 0\][/tex]
4. Divide by 150:
[tex]\[x = 0\][/tex]
This equation also has a single solution, [tex]\(x = 0\)[/tex].
---
Equation C: [tex]\(-75x + 57 = -75x + 57\)[/tex]
1. Simplify both sides:
[tex]\[-75x + 57 = -75x + 57\][/tex]
Since both sides are identical already, the equation is true for all values of [tex]\(x\)[/tex].
This equation has infinitely many solutions.
---
Equation D: [tex]\(57x + 57 = -75x - 75\)[/tex]
1. Add [tex]\(75x\)[/tex] to both sides to eliminate [tex]\(x\)[/tex] on the right:
[tex]\[57x + 75x + 57 = -75\][/tex]
2. Simplify the left side:
[tex]\[132x + 57 = -75\][/tex]
3. Subtract 57 from both sides:
[tex]\[132x = -132\][/tex]
4. Divide by 132:
[tex]\[x = -1\][/tex]
This equation has a single solution, [tex]\(x = -1\)[/tex].
---
Therefore, the equation with infinitely many solutions is (C) [tex]\(-75x + 57 = -75x + 57\)[/tex].
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