Answer :
To determine which of the given equations have infinitely many solutions, let's analyze each equation one by one:
1. Equation A: [tex]\(76x + 76 = -76x + 76\)[/tex]
This equation sets the left side equal to the right side with opposite coefficients for [tex]\(x\)[/tex].
Simplifying both sides:
[tex]\[
76x + 76 = -76x + 76
\][/tex]
Subtract [tex]\(76\)[/tex] from both sides:
[tex]\[
76x = -76x
\][/tex]
Adding [tex]\(76x\)[/tex] to both sides:
[tex]\[
152x = 0
\][/tex]
Dividing both sides by 152:
[tex]\[
x = 0
\][/tex]
Since there is a specific solution ([tex]\(x = 0\)[/tex]), it does not have infinitely many solutions.
2. Equation B: [tex]\(76x + 76 = 76x + 76\)[/tex]
This equation is identical on both sides:
[tex]\[
76x + 76 = 76x + 76
\][/tex]
Subtracting [tex]\(76x\)[/tex] and [tex]\(76\)[/tex] from both sides:
[tex]\[
0 = 0
\][/tex]
This is always true, meaning any value of [tex]\(x\)[/tex] satisfies the equation. Therefore, this equation has infinitely many solutions.
3. Equation C: [tex]\(-76x + 76 = -76x + 76\)[/tex]
Similarly, both sides of this equation are identical:
[tex]\[
-76x + 76 = -76x + 76
\][/tex]
Subtracting [tex]\(-76x\)[/tex] and [tex]\(76\)[/tex] from both sides:
[tex]\[
0 = 0
\][/tex]
This statement is always true; thus, this equation has infinitely many solutions as well.
4. Equation D: [tex]\(-76x + 76 = 76x + 76\)[/tex]
This equation sets the left side equal to the right side with opposite coefficients for [tex]\(x\)[/tex]:
Simplifying both sides:
[tex]\[
-76x + 76 = 76x + 76
\][/tex]
Subtract [tex]\(76\)[/tex] from both sides:
[tex]\[
-76x = 76x
\][/tex]
Adding [tex]\(76x\)[/tex] to both sides:
[tex]\[
0 = 152x
\][/tex]
Dividing by 152:
[tex]\[
x = 0
\][/tex]
Since it again results in a specific solution, it does not have infinitely many solutions.
In conclusion, the equations with infinitely many solutions are:
- (B) [tex]\(76x + 76 = 76x + 76\)[/tex]
- (C) [tex]\(-76x + 76 = -76x + 76\)[/tex]
1. Equation A: [tex]\(76x + 76 = -76x + 76\)[/tex]
This equation sets the left side equal to the right side with opposite coefficients for [tex]\(x\)[/tex].
Simplifying both sides:
[tex]\[
76x + 76 = -76x + 76
\][/tex]
Subtract [tex]\(76\)[/tex] from both sides:
[tex]\[
76x = -76x
\][/tex]
Adding [tex]\(76x\)[/tex] to both sides:
[tex]\[
152x = 0
\][/tex]
Dividing both sides by 152:
[tex]\[
x = 0
\][/tex]
Since there is a specific solution ([tex]\(x = 0\)[/tex]), it does not have infinitely many solutions.
2. Equation B: [tex]\(76x + 76 = 76x + 76\)[/tex]
This equation is identical on both sides:
[tex]\[
76x + 76 = 76x + 76
\][/tex]
Subtracting [tex]\(76x\)[/tex] and [tex]\(76\)[/tex] from both sides:
[tex]\[
0 = 0
\][/tex]
This is always true, meaning any value of [tex]\(x\)[/tex] satisfies the equation. Therefore, this equation has infinitely many solutions.
3. Equation C: [tex]\(-76x + 76 = -76x + 76\)[/tex]
Similarly, both sides of this equation are identical:
[tex]\[
-76x + 76 = -76x + 76
\][/tex]
Subtracting [tex]\(-76x\)[/tex] and [tex]\(76\)[/tex] from both sides:
[tex]\[
0 = 0
\][/tex]
This statement is always true; thus, this equation has infinitely many solutions as well.
4. Equation D: [tex]\(-76x + 76 = 76x + 76\)[/tex]
This equation sets the left side equal to the right side with opposite coefficients for [tex]\(x\)[/tex]:
Simplifying both sides:
[tex]\[
-76x + 76 = 76x + 76
\][/tex]
Subtract [tex]\(76\)[/tex] from both sides:
[tex]\[
-76x = 76x
\][/tex]
Adding [tex]\(76x\)[/tex] to both sides:
[tex]\[
0 = 152x
\][/tex]
Dividing by 152:
[tex]\[
x = 0
\][/tex]
Since it again results in a specific solution, it does not have infinitely many solutions.
In conclusion, the equations with infinitely many solutions are:
- (B) [tex]\(76x + 76 = 76x + 76\)[/tex]
- (C) [tex]\(-76x + 76 = -76x + 76\)[/tex]
