Question

For which of the following ordered pairs $(\mu ,\delta )$, the system of linear equations

$$\begin{gathered} x+2y+3z=1 \\ 3x+4y+5z=\mu \\ 4x+4y+4z=\delta \end{gathered}$$is inconsistent?

• A. $\left(4,6\right)$
• B. $\left(3,4\right)$
• C. $\left(1,0\right)$
• D. $\left(4,3\right)$

Answer 1

The correct option is D

$\left(4,3\right)$

The explanation for the correct option:

Step 1. Given a system of linear equations :

$$\begin{gathered} x+2y+3z=1 \\ 3x+4y+5z=\mu \\ 4x+4y+4z=\delta \end{gathered}$$

Now,

$\begin{array}{rcl}D& =& \left|\begin{array}{ccc}1& 2& 3\\ 3& 4& 5\\ 4& 4& 4\end{array}\right|\\ & =& 1(16-20)-2(12-20)+3(12-16)\\ & =& \left(-4\right)-2\left(-8\right)+3\left(-4\right)\\ & =& -4+16-12\\ & =& 0\end{array}$

we can write the equation as,

$$\begin{gathered} \left[\begin{array}{c}\begin{array}{ccc}1& 2& 3\\ 3& 4& 5\\ 4& 4& 4\end{array}\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}\begin{array}{c}1\\ \mu \\ \delta \end{array}\end{array}\right] \\ \left|\begin{array}{ccccc}1& 2& 3& :& 1\\ 3& 4& 5& :& \mu \\ 4& 4& 4& :& \delta \end{array}\right|\left[R_3\to R_3+2R_1-2R_2\right] \\ \left|\begin{array}{ccccc}1& 2& 3& :& 1\\ 3& 4& 5& :& \mu \\ 0& 0& 0& :& \delta -2\mu +2\end{array}\right| \end{gathered}$$

Step 2. Compare the third row:

By Compare third row, we get

$0x+0y+0z=\delta -2\mu +2$

$\Rightarrow$$\delta -2\mu +2=0$(the system will be consistent)

to make it inconsistent, $\delta -2\mu +2\ne 0$

Step 3. We will check with option which satisfy $(\mu ,\delta )$ values:

For $(\mu ,\delta )=\left(4,6\right)$

Now, put in $\delta -2\mu +2$

$\Rightarrow$$6-2*4+2=0$ (Consistent)

it does not satisfy the condition of inconsistency

For $(\mu ,\delta )=\left(3,4\right)$

Now, put in $\delta -2\mu +2$

$\Rightarrow$$4-2*3+2=0$ (Consistent)

it does not satisfy the condition of inconsistency

For $(\mu ,\delta )=\left(1,0\right)$

Now, put in $\delta -2\mu +2$

$\Rightarrow$$0-2*1+2=0$ (Consistent)

it does not satisfy the condition of inconsistency

For $(\mu ,\delta )=\left(4,3\right)$

Now, put in $\delta -2\mu +2$

$\Rightarrow$$3-2*4+2\ne 0$ (Inconsistent)

it does not satisfy the condition of inconsistency

Hence, Option ‘D’ is Correct.