Answer :
Final answer:
Options B (-76x+76=-76x+76) and C (76x + 76 = 76x +76) both simplify to a statement that is always true (0=0), indicating that they have infinitely many solutions. Option A has only one solution, and Option D is not an equation in x.
Explanation:
The question pertains to infinitely many solutions for equations. An equation has infinitely many solutions if it simplifies to a statement that is always true, such as 0 = 0.
- Option A (76x+ 76 = -76x+76) simplifies to 76x + 76x = 0, which results in 152x = 0. This means there's only one solution for x, which is 0. So option A does not have infinitely many solutions.
- Option B (-76x+76=-76x+76) simplifies straightforwardly to 0=0 because the terms on both sides of the equation cancel each other out. This means option B has infinitely many solutions because no matter what value x takes, the statement will always be true.
- Option C (76x + 76 = 76x +76) is similar to option B; simplifying it also leads to 0=0. Therefore, option C also has infinitely many solutions.
- Option D (-76x + 76x +76) simplifies to 0 + 76, which is not an equation but rather a statement that 76 equals 76, which is always true. However, since it does not involve x, it cannot be considered to have infinitely many solutions in the context of solving for x.
Options B and C have infinitely many solutions.
Final answer:
Equations B: -76x+76=-76x+76 and C: 76x + 76 = 76x +76 both have infinitely many solutions as they can be simplified to 0 = 0. Equation A: 76x+ 76 = -76x+76 does not have infinitely many solutions because it simplifies to a non-trivial statement.
Explanation:
The question relates to determining which equations have infinitely many solutions. An equation has infinitely many solutions if it can be simplified to a trivial truth, such as a statement where the same value or expression is on both sides of the equals sign.
A: 76x+ 76 = -76x+76 does not have infinitely many solutions because simplifying it gives 76x + 76x = 0, indicating that x can have only one specific value to satisfy the equation.
B: -76x+76=-76x+76 does have infinitely many solutions. Simplifying this gives 0 = 0, a trivial truth regardless of the value of x.
C: 76x + 76 = 76x +76 does have infinitely many solutions. Simplifying leads to 0 = 0, which is always true for any value of x.
D: -76x + 76x +76 is not an equation since it is missing an equals sign and therefore we cannot determine if it has solutions without additional information.
Solution B and C are both correct as they can be simplified to statements that are always true, indicating that they have infinitely many solutions.
