Certainly! Let's analyze each equation step-by-step to determine which one has an infinite number of solutions.
### Equation I:
[tex]\[ 4 - 2x = 2(2 - x) \][/tex]
Let's simplify it.
1. Distribute the 2 on the right-hand side:
[tex]\[ 4 - 2x = 4 - 2x \][/tex]
2. Notice that both sides of the equation are identical:
[tex]\[ 4 - 2x = 4 - 2x \][/tex]
Since the equation is true for all values of [tex]\( x \)[/tex], this equation has an infinite number of solutions.
### Equation II:
[tex]\[ 4x - 3 = 2x - 5 \][/tex]
Let's simplify it.
1. Subtract [tex]\( 2x \)[/tex] from both sides:
[tex]\[ 4x - 2x - 3 = -5 \][/tex]
[tex]\[ 2x - 3 = -5 \][/tex]
2. Add 3 to both sides:
[tex]\[ 2x - 3 + 3 = -5 + 3 \][/tex]
[tex]\[ 2x = -2 \][/tex]
3. Divide by 2:
[tex]\[ x = -1 \][/tex]
This equation has a single unique solution: [tex]\( x = -1 \)[/tex].
### Equation III:
[tex]\[ 8x = 5x + 3 \][/tex]
Let's simplify it.
1. Subtract [tex]\( 5x \)[/tex] from both sides:
[tex]\[ 8x - 5x = 3 \][/tex]
[tex]\[ 3x = 3 \][/tex]
2. Divide by 3:
[tex]\[ x = 1 \][/tex]
This equation also has a single unique solution: [tex]\( x = 1 \)[/tex].
### Equation IV:
[tex]\[ 12 - 3x = 6 - 3x \][/tex]
Let's simplify it.
1. Add [tex]\( 3x \)[/tex] to both sides:
[tex]\[ 12 - 3x + 3x = 6 - 3x + 3x \][/tex]
[tex]\[ 12 = 6 \][/tex]
This is a contradiction because 12 is not equal to 6. Thus, this equation has no solutions.
### Conclusion:
Among the given equations, Equation I:
[tex]\[ 4 - 2x = 2(2 - x) \][/tex]
has an infinite number of solutions.
So, the answer is:
I
