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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.
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.
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