High School

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 equations have infinitely many solutions, we need to look for cases where both sides of the equation are identical for any value of [tex]\( x \)[/tex]. Let's analyze each equation one by one:

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

Subtract [tex]\( 76x \)[/tex] from both sides:
[tex]\[ 76x + 76 - 76x = -76x + 76 - 76x \][/tex]
This simplifies to:
[tex]\[ 76 = -152x + 76 \][/tex]
Subtracting 76 from both sides gives:
[tex]\[ 0 = -152x \][/tex]
This equation has a unique solution, not infinitely many solutions.

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

Both sides are identical. Since the left-hand side and the right-hand side are the exact same expressions, the equation holds true for all values of [tex]\( x \)[/tex]. Hence, this equation has infinitely many solutions.

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

Subtract 76 from both sides:
[tex]\[ -76x + 76 - 76 = 76x + 76 - 76 \][/tex]
Which simplifies to:
[tex]\[ -76x = 76x \][/tex]
Add [tex]\( 76x \)[/tex] to both sides:
[tex]\[ 0 = 152x \][/tex]
This equation has a unique solution, not infinitely many solutions.

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

Both sides are identical. Since the left-hand side and the right-hand side are the exact same expressions, this equation is true for any value of [tex]\( x \)[/tex]. Hence, this equation has infinitely many solutions.

Based on this analysis, the equations that have infinitely many solutions are:

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

Thus, the correct answers are B and D.