Question

Find the value of k for which the system of equations
8x + 5y = 9, kx + 10y = 15
has a non-zero solution.

Answer 1

The given system of equations:
8x + 5y = 9
8x + 5y - 9 = 0                   ….(i)
kx + 10y = 15
kx + 10y - 15 = 0                  ….(ii)
These equations are of the following form:
`a_1x+b_1y+c_1 = 0, a_2x+b_2y+c_2 = 0`
where, `a_1 = 8, b_1= 5, c_1= -9 and a_2 = k, b_2 = 10, c_2= – 15`
In order that the given system has no solution, we must have:
`(a_1)/(a_2) = (b_1)/(b_2) ≠ (c_1)/(c_2)`
` i.e.,  8/k = 5/10 ≠( −9)/(−15)`
`i.e. , 8/k = 1/2 ≠ 3/5`
`8/k = 1/2 and 8/k ≠ 3/5`
`⇒ k = 16 and k ≠ 40/3`
Hence, the given system of equations has no solutions when k is equal to 16.