Find the values of N for which the solution of the equations 3x+2y=Nx and 15x+4y=Nx is not unique
To find the values of N for which the solution of the given system of equations is not unique, we need to determine when the two equations are dependent or have infinitely many solutions.
To do this, let's start by simplifying the equations:
Equation 1: 3x + 2y = Nx
Equation 2: 15x + 4y = Nx
To determine if the equations are dependent, we need to compare their slopes. The slopes can be found by rearranging each equation into slope-intercept form (y = mx + b) where 'm' is the slope:
Equation 1: 2y = (N - 3)x => y = (N - 3)x / 2 (slope: (N - 3) / 2)
Equation 2: 4y = (N - 15)x => y = (N - 15)x / 4 (slope: (N - 15) / 4)
For the equations to be dependent, the slopes should be equal. Therefore, we need to set the two slopes equal to each other and solve for N:
(N - 3) / 2 = (N - 15) / 4
To solve this equation, we'll cross-multiply:
4(N - 3) = 2(N - 15)
4N - 12 = 2N - 30
2N = -18
N = -9
Therefore, when N = -9, the solution of the equations will not be unique. In other words, the system will have infinitely many solutions. For any other value of N, the solution will be unique.