Final answer:
The pair of equations that represents an inconsistent system of linear equations is Option D: 5y - x = -10 and 4x - 10y = -24. This is because the first equation, when multiplied by -4, results in the same coefficients for x and y as the second equation but with a different constant, indicating the lines are parallel.
Step-by-step explanation:
The question asks which pair of equations represents an inconsistent system of linear equations. An inconsistent system has no solution, which occurs when the two lines are parallel and therefore have the same slope but different y-intercepts. To find the inconsistent system, we must look for a pair of equations where the coefficients of x and y in one equation are multiples of the coefficients in the other, but the constants are not in the same proportion.
Option A:
2x + 5y = 12
2x - 10y = -5
These equations are not multiples of each other.
Option B:
2x + 5y = 12
4x - 10y = -24
This option shows equations that when the first is multiplied by two, it does not result in the second equation.
Option C:
5y - x = -10
2x - 10y = -5
These are also not multiples of each other.
Option D:
5y - x = -10
4x - 10y = -24
This pair is the correct answer because if we multiply the first equation by -4, we get:
-20y + 4x = 40,
which has the same coefficients for x and y as the second equation, but a different constant term, indicating parallel lines and thus an inconsistent system.