Answer :
To determine which equations have infinitely many solutions, we need to analyze each equation to see when both sides are exactly the same for any value of [tex]\(x\)[/tex].
Let's examine each equation one by one:
Equation A: [tex]\(76x + 76 = 76x + 76\)[/tex]
- Both sides of the equation are exactly the same.
- This means that for any value of [tex]\(x\)[/tex], the equation will hold true since each side is identical.
- Hence, this equation has infinitely many solutions.
Equation B: [tex]\(76x + 76 = -76x + 76\)[/tex]
- If we simplify both sides, we get:
- [tex]\(76x + 76 = -76x + 76\)[/tex]
- Bring like terms involving [tex]\(x\)[/tex] together: [tex]\(76x + 76x = 0\)[/tex]
- Simplifying gives [tex]\(152x = 0\)[/tex]
- Solving for [tex]\(x\)[/tex], we find [tex]\(x = 0\)[/tex].
- This equation has exactly one solution, which is [tex]\(x = 0\)[/tex].
Equation C: [tex]\(-76x + 76 = 76x + 76\)[/tex]
- Simplifying both sides, we get:
- [tex]\(-76x + 76 = 76x + 76\)[/tex]
- Bring like terms involving [tex]\(x\)[/tex] together: [tex]\(-76x - 76x = 0\)[/tex]
- Simplifying gives [tex]\(-152x = 0\)[/tex]
- Solving for [tex]\(x\)[/tex], we find [tex]\(x = 0\)[/tex].
- This equation has exactly one solution, which is [tex]\(x = 0\)[/tex].
Equation D: [tex]\(-76x + 76 = -76x + 76\)[/tex]
- Both sides of the equation are exactly the same.
- This implies that for any value of [tex]\(x\)[/tex], the equation will hold true since each side is identical.
- Therefore, this equation has infinitely many solutions.
Conclusion:
Equations A and D have infinitely many solutions because they are true for any value of [tex]\(x\)[/tex].
Let's examine each equation one by one:
Equation A: [tex]\(76x + 76 = 76x + 76\)[/tex]
- Both sides of the equation are exactly the same.
- This means that for any value of [tex]\(x\)[/tex], the equation will hold true since each side is identical.
- Hence, this equation has infinitely many solutions.
Equation B: [tex]\(76x + 76 = -76x + 76\)[/tex]
- If we simplify both sides, we get:
- [tex]\(76x + 76 = -76x + 76\)[/tex]
- Bring like terms involving [tex]\(x\)[/tex] together: [tex]\(76x + 76x = 0\)[/tex]
- Simplifying gives [tex]\(152x = 0\)[/tex]
- Solving for [tex]\(x\)[/tex], we find [tex]\(x = 0\)[/tex].
- This equation has exactly one solution, which is [tex]\(x = 0\)[/tex].
Equation C: [tex]\(-76x + 76 = 76x + 76\)[/tex]
- Simplifying both sides, we get:
- [tex]\(-76x + 76 = 76x + 76\)[/tex]
- Bring like terms involving [tex]\(x\)[/tex] together: [tex]\(-76x - 76x = 0\)[/tex]
- Simplifying gives [tex]\(-152x = 0\)[/tex]
- Solving for [tex]\(x\)[/tex], we find [tex]\(x = 0\)[/tex].
- This equation has exactly one solution, which is [tex]\(x = 0\)[/tex].
Equation D: [tex]\(-76x + 76 = -76x + 76\)[/tex]
- Both sides of the equation are exactly the same.
- This implies that for any value of [tex]\(x\)[/tex], the equation will hold true since each side is identical.
- Therefore, this equation has infinitely many solutions.
Conclusion:
Equations A and D have infinitely many solutions because they are true for any value of [tex]\(x\)[/tex].
