Question

Find the value of k for the pair of linear equations 4x-5y=5 and kx+3y=3 is inconsistent.

Answer 1

Hint: First, before proceeding for this, we must know the following condition for the linear equations to be inconsistent or has no solution or they are parallel lines. Then, the condition for the linear equations ${{a}_{1}}x+{{b}_{1}}y={{c}_{1}}$and ${{a}_{2}}x+{{b}_{2}}y={{c}_{2}}$must be inconsistent if it has no solution given by the condition as$\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{b}_{1}}}{{{b}_{2}}}\ne \dfrac{{{c}_{1}}}{{{c}_{2}}}$. Then, by substituting all the values of constants in the condition of inconsistency and by using the first two equality relation, we get the value of k.

Complete step by step answer:
In this question, we are supposed to find the value of k for the pair of linear equations 4x-5y=5 and kx+3y=3 is inconsistent.
So, before proceeding for this, we must know the following condition for the linear equations to be inconsistent or has no solution or they are parallel lines.
Now, the condition for the linear equations ${{a}_{1}}x+{{b}_{1}}y={{c}_{1}}$and ${{a}_{2}}x+{{b}_{2}}y={{c}_{2}}$must be inconsistent if it has no solution given by the condition as:
$\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{b}_{1}}}{{{b}_{2}}}\ne \dfrac{{{c}_{1}}}{{{c}_{2}}}$
Now, we have the equations from the question as 4x-5y=5 and kx+3y=3 which gives the values of the constants as:
${{a}_{1}}=4,{{a}_{2}}=k,{{b}_{1}}=-5,{{b}_{2}}=3,{{c}_{1}}=5$and ${{c}_{2}}=3$
Now, by substituting all the values of constants in the condition of inconsistency, we get:
$\dfrac{4}{k}=\dfrac{-5}{3}\ne \dfrac{5}{3}$
So, by using the first two equality relation, we get the value of k as:
$\begin{align}
  & \dfrac{4}{k}=\dfrac{-5}{3} \\
 & \Rightarrow 4\times 3=-5\times k \\
 & \Rightarrow k=\dfrac{-12}{5} \\
\end{align}$

Hence, we get the value of k as $\dfrac{-12}{5}$for the equations to be inconsistent which is the required result.

Note: Now, to solve these type of the questions we can also use the approach of augmented matrix form $\left[ A\left| B \right. \right]$ in which equations are represented where $\rho \left( A\left| B \right. \right)$ represents the rank of augmented matrix. So, for no solution, we have the condition to be fulfilled where $\rho \left( A \right)$is rank of matrix A which is formed by the coefficients of the linear equations and B is the matrix of constant terms as:
$\rho \left( A \right)\ne \rho \left( A\left| B \right. \right)$.