Question

For what values of $k$ is the linear system consistent?

$\left\{\begin{array}{l}6x_1-5x_2=4\\ 9x_1+k{x}_2=-1\end{array}\right.$

$\left\{\begin{array}{l}6x_1-8x_2=k\\ -9x_1+12x_2=-1\end{array}\right.$

Answer 1

Step-1:Consistent:

The system of linear equation be $a_{1}x+b_{1}y=c_1;a_{2}x+b_{2}y=c_2$. it can be written in matrix form as

$\left[\begin{array}{cc}{a}_1& b_1\\ a_2& b_2\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}{c}_1\\ c_2\end{array}\right]$ which is represented as $AX=C$

if the determined of $A$ is equal to zero then the system is consistent.

Step-2: Finding the value of $k$:

Given,

$\left\{\begin{array}{l}6x_1-5x_2=4\\ 9x_1+k{x}_2=-1\end{array}\right.$is consistent

So,

$$\begin{gathered} \Rightarrow \left|A\right|\ne 0 \\ \Rightarrow \left|\begin{array}{cc}6& -5\\ 9& k\end{array}\right|\ne 0 \\ \Rightarrow 6k+45\ne 0 \\ \Rightarrow 6k\ne -45 \\ \Rightarrow k\ne \frac{-45}{6} \\ \Rightarrow k\ne \frac{-15}{3} \end{gathered}$$

Hence, when $k\in R-\left\{\frac{-15}{3}\right\}$ the linear system of equation $\left\{\begin{array}{l}6x_1-5x_2=4\\ 9x_1+k{x}_2=-1\end{array}\right.$ is consistent.

Step-3: Finding the value of $k$ for other system:

Given,

$\left\{\begin{array}{l}6x_1-8x_2=k\\ -9x_1+12x_2=-1\end{array}\right.$is consistent

when

$\Rightarrow \left|A\right|\ne 0;\left|{∆}_1\right|\ne 0\text{and}\left|{∆}_1\right|\ne 0$

Here $\Rightarrow \left|A\right|\ne 0$

So we check,

$$\begin{gathered} \Rightarrow \left|{∆}_1\right|\ne 0 \\ \Rightarrow \left|\begin{array}{cc}k& -8\\ -1& 12\end{array}\right|\ne 0 \\ \Rightarrow 12k-8\ne 0 \\ \Rightarrow 12k\ne 8 \\ \Rightarrow k\ne \frac{8}{12} \\ \Rightarrow k\ne \frac{2}{3} \end{gathered}$$

$$\begin{gathered} \Rightarrow \left|{∆}_2\right|\ne 0 \\ \Rightarrow \left|\begin{array}{cc}6& k\\ -9& -1\end{array}\right|\ne 0 \\ \Rightarrow -6+9k\ne 0 \\ \Rightarrow 9k\ne 6 \\ \Rightarrow k\ne \frac{6}{9} \\ \Rightarrow k\ne \frac{2}{3} \end{gathered}$$

Hence, when $k\in R-\left\{\frac{-15}{3}\right\}$ the linear system of equation $\left\{\begin{array}{l}6x_1-5x_2=4\\ 9x_1+k{x}_2=-1\end{array}\right.$ is consistent.

And when $k\in R-\left\{\frac{2}{3}\right\}$ the linear system of equation $\left\{\begin{array}{l}6x_1-8x_2=k\\ -9x_1+12x_2=-1\end{array}\right.$is consistent.