Question

for what value of k the following system of eq. have unique solution and infinetely many solutions.

please solve the question step by step........​

Answer 1

Information:

For the system of equations a₁x+b₁y=c and a₂x+b₂y=d,

• Condition to have a unique solution is

$\longrightarrow\dfrac{a_{1}}{a_{2}}\neq\dfrac{b_{1}}{b_{2}}$

• Condition to have infinitely many solutions is

$\longrightarrow\dfrac{a_{1}}{a_{2}}=\dfrac{b_{1}}{b_{2}}=\dfrac{c}{d}$

1)

i) Given:

Two equations-

• 2x+ky=1
• 3x-5y=7

To Find:

• Value of k for which given system of equation have a unique solution

Solution:

On applying condition for unique solution, we get

$\longrightarrow\sf{\dfrac{2}{3}\neq\dfrac{k}{-5}}$

$\longrightarrow\sf{\dfrac{k}{-5}\neq\dfrac{2}{3}}$

$\longrightarrow\sf{k\neq\dfrac{-10}{3}}$

ii) Given:

Two equations-

• x-2y=3
• 3x+ky=1

To Find:

• Value of k for which given system of equation have a unique solution

Solution:

On applying condition for unique solution, we get

$\longrightarrow\sf{\dfrac{1}{3}\neq\dfrac{-2}{k}}$

$\longrightarrow\sf{k\neq-6}$

iii) Given:

Two equations-

• 2x+5y=7
• 3x-ky=5

To Find:

• Value of k for which given system of equation have a unique solution

Solution:

On applying condition for unique solution, we get

$\longrightarrow\sf{\dfrac{2}{3}\neq\dfrac{5}{-k}}$

$\longrightarrow\sf{k\neq\dfrac{-15}{2}}$

2)

i) Given:

Two equations-

• 7x-y=5
• 21x-3y=k

To Find:

• Value of k for which given system of equation have infinitely many solutions

Solution:

On applying condition for infinitely many solutions, we get

$\longrightarrow\sf{\dfrac{7}{21}=\dfrac{-1}{-3}=\dfrac{5}{k}}$

$\longrightarrow\sf{\dfrac{1}{3}=\dfrac{1}{3}=\dfrac{5}{k}}$

So,

$\longrightarrow\sf{\dfrac{1}{3}=\dfrac{5}{k}}$

$\longrightarrow\sf{k=15}$

ii) Given:

Two equations-

• 5x+2y=k
• 10x+4y=3

To Find:

• Value of k for which given system of equation have infinitely many solutions

Solution:

On applying condition for infinitely many solutions, we get

$\longrightarrow\sf{\dfrac{5}{10}=\dfrac{2}{4}=\dfrac{k}{3}}$

$\longrightarrow\sf{\dfrac{1}{2}=\dfrac{1}{2}=\dfrac{k}{3}}$

So,

$\longrightarrow\sf{\dfrac{1}{2}=\dfrac{k}{3}}$

$\longrightarrow\sf{\dfrac{k}{3}=\dfrac{1}{2}}$

$\longrightarrow\sf{k=\dfrac{3}{2}}$

iii) Given:

Two equations-

• kx+4y=k-4
• 16x+ky=k

To Find:

• Value of k for which given system of equation have infinitely many solutions

Solution:

On applying condition for infinitely many solutions, we get

$\longrightarrow\sf{\dfrac{k}{16}=\dfrac{4}{k}=\dfrac{k-4}{k}}$

So,

$\longrightarrow\sf{\dfrac{k}{16}=\dfrac{4}{k}}$

$\longrightarrow\sf{k^{2}=64}$

$\longrightarrow\sf{k=\sqrt{64}}$

$\longrightarrow\sf{k=\pm8}$

$\longrightarrow\sf{k=8,-8}$