Which equation has exactly ONE solution?
A. 3+10x/2 = 4x + 2
B. 15x−5/3 = 5x + 4
C. 2+6x2 = 3x + 1
D. 4x−10/2 = 2x − 5
To determine which equation has exactly one solution, we need to find the equation that, once simplified, results in a linear equation.
Let's simplify each of the given equations:
A. 3 + 10x/2 = 4x + 2
Multiplying through by 2, we get:
6 + 10x = 8x + 4
Subtracting 6 and 8x from both sides, we have:
10x - 8x = 4 - 6
2x = -2
Dividing by 2, we find:
x = -1
Therefore, equation A has exactly one solution.
B. 15x - 5/3 = 5x + 4
Multiplying through by 3, we get rid of the fraction:
45x - 5 = 15x + 12
Subtracting 15x and adding 5 from both sides, we obtain:
45x - 15x = 12 + 5
30x = 17
Dividing by 30, we find:
x = 17/30
Therefore, equation B does not have exactly one solution.
C. 2 + 6x^2 = 3x + 1
Rearranging the terms and simplifying, we have:
6x^2 - 3x + 2 - 1 = 0
6x^2 - 3x + 1 = 0
This is a quadratic equation, not a linear equation. Quadratic equations can have zero solutions, one solution, or two solutions. Since it does not state that this quadratic equation has exactly one solution, we cannot determine whether it does based on the given information.
D. 4x - 10/2 = 2x - 5
Multiplying through by 2 and simplifying the fractions, we get:
8x - 5 = 4x - 5
Subtracting 4x and adding 5 to both sides, we get:
8x - 4x = -5 + 5
4x = 0
Dividing by 4, we find:
x = 0
Therefore, equation D has exactly one solution.
In conclusion, the equations that have exactly one solution are A and D.