Question

The system of equations 2x – ay = 4; 5x – ky = 5 is inconsistent. If k and a are natural numbers, then the minimum possible value of (a + k) is equal to?

Answer 1

The minimum value of a + k is 7

Step-by-step explanation:

The given linear equations

$2x-ay=4$

$5x-ky=5$

we know that if two simultaneous linear equations $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$ is inconsistent

Then

$\frac{a_1}{a_2}=\frac{b_1}{b_2}$

Therefore, here

$\frac{2}{5}=\frac{-a}{-k}$

$\implies 2k=5a$

$\implies a=\frac{2k}{5}$

Now,

$a+k=\frac{2k}{5}+k=\frac{7k}{5}$

Since a and k are natural numbers

Therefore, a + k must be a natural number

i.e. k must be multiple of 5

i.e. k = 5, 10, 15, 20, ....

Minimum value of a + k will be for k = 5

Thus, the minimum value of a + k

$a+k=\frac{7\times 5}{5}=7$

Hope this answer is helpful.

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