Which of the following equations has an infinite number of solutions?
A. 3x-3=-4x
B. 2y+4-y=16
C. 7x+5=4x+5+3x
D. 6y-2=2(y-1)***
Is the answer D?
No, D has the solution y=0
6y-2 = 2y-2
4y = 0
y=0
C is the one you want:
7x+5 = 7x+5
Both sides represent the same value, so any solution of the left side is also a solution on the right side.
Yes, the correct answer is D.
To determine which equation has an infinite number of solutions, we need to examine the equations and see if they lead to a contradiction or if they are identically true. A contradiction would mean that the equation has no solutions, while being identically true would mean that any value of the variable(s) would satisfy the equation.
Let's analyze each equation one by one:
A. 3x - 3 = -4x
To solve this equation, we can simplify it by combining like terms:
3x + 4x = 3
7x = 3
x = 3/7
This equation has a single solution, so it does not have an infinite number of solutions.
B. 2y + 4 - y = 16
Simplifying this equation gives:
y + 4 = 16
y = 16 - 4
y = 12
Again, we obtain a single solution, so this equation does not have an infinite number of solutions.
C. 7x + 5 = 4x + 5 + 3x
Simplifying and combining like terms leads to:
7x - 4x - 3x = 5 - 5
0 = 0
This equation is identically true since 0 is equal to 0. Therefore, any value of x would satisfy this equation, resulting in an infinite number of solutions.
D. 6y - 2 = 2(y - 1)
Expanding and simplifying the equation gives:
6y - 2 = 2y - 2
6y - 2y = -2 + 2
4y = 0
Dividing both sides of the equation by 4, we get:
y = 0
Once again, we have a single solution, indicating that this equation does not possess an infinite number of solutions.
Based on our analysis, the equation with an infinite number of solutions is C, not D.