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Answer :
To determine which of the given equations have infinitely many solutions, let's analyze each one step by step:
An equation has infinitely many solutions when both sides of the equation are identical for all values of the variable. This means that after simplifying both sides, we end up with a true identity like [tex]\(0 = 0\)[/tex].
Let's go through each option:
(A) [tex]\(57x + 57 = -75x - 75\)[/tex]
1. Combine like terms:
[tex]\(57x + 75x = -57 - 75\)[/tex]
[tex]\(132x = -132\)[/tex]
2. This reduces to:
[tex]\(x = -1\)[/tex]
This equation has only one solution, [tex]\(x = -1\)[/tex], not infinitely many.
(B) [tex]\(-75x + 57 = -75x + 57\)[/tex]
1. Notice both sides are exactly the same:
[tex]\(-75x + 57 = -75x + 57\)[/tex]
2. This is always true, no matter the value of [tex]\(x\)[/tex].
Therefore, this equation has infinitely many solutions.
(C) [tex]\(75x + 57 = -75x + 57\)[/tex]
1. Combine like terms:
[tex]\(75x + 75x = 0\)[/tex]
[tex]\(150x = 0\)[/tex]
2. This simplifies to:
[tex]\(x = 0\)[/tex]
This equation has only one solution, [tex]\(x = 0\)[/tex], not infinitely many.
(D) [tex]\(-57x + 57 = -75x + 75\)[/tex]
1. Rearrange and simplify:
[tex]\(-57x + 75x = 75 - 57\)[/tex]
[tex]\(18x = 18\)[/tex]
2. This simplifies to:
[tex]\(x = 1\)[/tex]
This equation has only one solution, [tex]\(x = 1\)[/tex], not infinitely many.
Based on this analysis, the equation with infinitely many solutions is:
- (B) [tex]\(-75x + 57 = -75x + 57\)[/tex]
Therefore, only option (B) has infinitely many solutions.
An equation has infinitely many solutions when both sides of the equation are identical for all values of the variable. This means that after simplifying both sides, we end up with a true identity like [tex]\(0 = 0\)[/tex].
Let's go through each option:
(A) [tex]\(57x + 57 = -75x - 75\)[/tex]
1. Combine like terms:
[tex]\(57x + 75x = -57 - 75\)[/tex]
[tex]\(132x = -132\)[/tex]
2. This reduces to:
[tex]\(x = -1\)[/tex]
This equation has only one solution, [tex]\(x = -1\)[/tex], not infinitely many.
(B) [tex]\(-75x + 57 = -75x + 57\)[/tex]
1. Notice both sides are exactly the same:
[tex]\(-75x + 57 = -75x + 57\)[/tex]
2. This is always true, no matter the value of [tex]\(x\)[/tex].
Therefore, this equation has infinitely many solutions.
(C) [tex]\(75x + 57 = -75x + 57\)[/tex]
1. Combine like terms:
[tex]\(75x + 75x = 0\)[/tex]
[tex]\(150x = 0\)[/tex]
2. This simplifies to:
[tex]\(x = 0\)[/tex]
This equation has only one solution, [tex]\(x = 0\)[/tex], not infinitely many.
(D) [tex]\(-57x + 57 = -75x + 75\)[/tex]
1. Rearrange and simplify:
[tex]\(-57x + 75x = 75 - 57\)[/tex]
[tex]\(18x = 18\)[/tex]
2. This simplifies to:
[tex]\(x = 1\)[/tex]
This equation has only one solution, [tex]\(x = 1\)[/tex], not infinitely many.
Based on this analysis, the equation with infinitely many solutions is:
- (B) [tex]\(-75x + 57 = -75x + 57\)[/tex]
Therefore, only option (B) has infinitely many solutions.
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