Answer :
To determine which equations have infinitely many solutions, we need to analyze each one and see if it holds true for all values of the variable [tex]\( x \)[/tex].
Equation (A): [tex]\( 75x + 57 = -75x + 57 \)[/tex]
- Combine like terms by moving terms with [tex]\( x \)[/tex] to one side:
[tex]\[
75x + 75x = 57 - 57
\][/tex]
This simplifies to:
[tex]\[
150x = 0
\][/tex]
Solving for [tex]\( x \)[/tex] gives:
[tex]\[
x = 0
\][/tex]
Therefore, equation (A) has one solution, not infinitely many.
Equation (B): [tex]\( 57x + 57 = -75x - 75 \)[/tex]
- Move terms with [tex]\( x \)[/tex] to one side:
[tex]\[
57x + 75x = -75 - 57
\][/tex]
This simplifies to:
[tex]\[
132x = -132
\][/tex]
Solving for [tex]\( x \)[/tex] gives:
[tex]\[
x = -1
\][/tex]
Therefore, equation (B) has one solution, not infinitely many.
Equation (C): [tex]\( -75x + 57 = -75x + 57 \)[/tex]
- Both sides of this equation are identical, which means every value of [tex]\( x \)[/tex] will satisfy this equation.
Therefore, this equation has infinitely many solutions.
Equation (D): [tex]\( -57x + 57 = -75x + 75 \)[/tex]
- Move terms with [tex]\( x \)[/tex] to one side:
[tex]\[
-57x + 75x = 75 - 57
\][/tex]
This simplifies to:
[tex]\[
18x = 18
\][/tex]
Solving for [tex]\( x \)[/tex] gives:
[tex]\[
x = 1
\][/tex]
Therefore, equation (D) has one solution, not infinitely many.
In summary, only Equation (C) has infinitely many solutions.
Equation (A): [tex]\( 75x + 57 = -75x + 57 \)[/tex]
- Combine like terms by moving terms with [tex]\( x \)[/tex] to one side:
[tex]\[
75x + 75x = 57 - 57
\][/tex]
This simplifies to:
[tex]\[
150x = 0
\][/tex]
Solving for [tex]\( x \)[/tex] gives:
[tex]\[
x = 0
\][/tex]
Therefore, equation (A) has one solution, not infinitely many.
Equation (B): [tex]\( 57x + 57 = -75x - 75 \)[/tex]
- Move terms with [tex]\( x \)[/tex] to one side:
[tex]\[
57x + 75x = -75 - 57
\][/tex]
This simplifies to:
[tex]\[
132x = -132
\][/tex]
Solving for [tex]\( x \)[/tex] gives:
[tex]\[
x = -1
\][/tex]
Therefore, equation (B) has one solution, not infinitely many.
Equation (C): [tex]\( -75x + 57 = -75x + 57 \)[/tex]
- Both sides of this equation are identical, which means every value of [tex]\( x \)[/tex] will satisfy this equation.
Therefore, this equation has infinitely many solutions.
Equation (D): [tex]\( -57x + 57 = -75x + 75 \)[/tex]
- Move terms with [tex]\( x \)[/tex] to one side:
[tex]\[
-57x + 75x = 75 - 57
\][/tex]
This simplifies to:
[tex]\[
18x = 18
\][/tex]
Solving for [tex]\( x \)[/tex] gives:
[tex]\[
x = 1
\][/tex]
Therefore, equation (D) has one solution, not infinitely many.
In summary, only Equation (C) has infinitely many solutions.
