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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, we will examine each equation:

(A) [tex]\(-76x + 76 = 76x + 76\)[/tex]

1. Start by moving all the terms involving [tex]\(x\)[/tex] to one side of the equation:
[tex]\(-76x - 76x = 76 - 76\)[/tex]

2. Combine like terms:
[tex]\(-152x = 0\)[/tex]

3. Divide both sides by [tex]\(-152\)[/tex]:
[tex]\(x = 0\)[/tex]

For equation (A), if you substitute [tex]\(x = 0\)[/tex], it satisfies the equation, indicating that [tex]\(x = 0\)[/tex] is a solution. Since the form suggests [tex]\(0 = 0\)[/tex] for any [tex]\(x\)[/tex], the equation has infinitely many solutions.

(B) [tex]\(76x + 76 = -76x + 76\)[/tex]

1. Move all terms involving [tex]\(x\)[/tex] to one side:
[tex]\(76x + 76x = 76 - 76\)[/tex]

2. Simplify:
[tex]\(152x = 0\)[/tex]

3. Divide both sides by 152:
[tex]\(x = 0\)[/tex]

Equation (B), like equation (A), also leads to [tex]\(x = 0\)[/tex] being a valid solution, but the structural form [tex]\(0 = 0\)[/tex] implies infinitely many solutions.

(C) [tex]\(-76x + 76 = -76x + 76\)[/tex]

1. The equation is already simplified, showing that both sides of the equation are equal regardless of [tex]\(x\)[/tex].

2. This essentially means it forms [tex]\(0 = 0\)[/tex], which is always true.

For equation (C), this condition implies that it has infinitely many solutions because any value of [tex]\(x\)[/tex] satisfies the equation.

(D) [tex]\(76x + 76 = 76x + 76\)[/tex]

1. Simplify both sides:
The equation essentially shows that the two sides are always equal.

2. This can be written as [tex]\(0 = 0\)[/tex], which is always true.

Thus, for equation (D), any value of [tex]\(x\)[/tex] will satisfy the equation, indicating infinitely many solutions.

In conclusion, all the given equations (A, B, C, and D) result in structural forms that hold true for any value of [tex]\(x\)[/tex], which means each one has infinitely many solutions.

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