Question:

Find the value of k for which the pair of equations kx + 2y = 3 and 3x + 6y = 10 has a unique solution.

Solution:

Given pair of linear equations is

kx + 2y = 3 …….(i)

3x + 6y = 10 …….(ii)

$a_1 = k,b_1 = 2, c_1 = 3$

$a_2 = 3, b_2 = 6, c_2 = 10$

For unique solution,

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

$\Rightarrow \frac{k}{3} \neq \frac{2}{6}$

$\Rightarrow k \neq 1$

Hence, the pair of equations has a unique solution for all real values of k except 1.

Find c if the system of equations cx + 3y + (3-c) = 0; 12x + cy – c = 0 has infinitely many solution.

Given system of equations is:

cx + 3y + (3-c) = 0 ………..(i)

12x + cy – c = 0 ………..(ii)

$a_1 = c,b_1 = 3, c_1 = 3-c$

$a_2 = 12, b_2 = c, c_2 = -c$

If the system of equation have infinitly many solution:

$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

$\Rightarrow \frac{c}{12}=\frac{3}{c}=\frac{3-c}{-c}$

$\Rightarrow c^2 = 36$

$\Rightarrow c = \pm 6$ ……(iii)

Also, $-3c = 3c - c^2$

$\Rightarrow c^2 - 6c = 0$

$\Rightarrow c(c - 6) = 0$

$\Rightarrow c = 0, 6$ ……..(iv)

From (iii) and (iv)

The value of c = 6