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Which of the following equations have infinitely many solutions? Choose all answers that apply:

A. [tex]76x + 76 = -76x + 76[/tex]

B. [tex]76x + 76 = 76x + 76[/tex]

C. [tex]-76x + 76 = -76x + 76[/tex]

D. [tex]-76x + 76 = 76x + 76[/tex]

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]

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