### <Success>Linear inequalities lecture-5</Success> <div style='float: left; width:99%;font-size:30px;text-align:left'> </div> <div style='float: right; width:1%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-05-linear-inequalities-l-5-6-a6fxrbgnlbo-1.jpg"></p>

### <Success>Example 1</Success> <div style='float: left; width:99%;font-size:20px;text-align:left'> <p>$ 2 x-4 y+z=3 $</p> <p>$ x-3 y+z=5 $</p> <p>$ 3 x-7 y+2 z=12 $</p> <p><strong>Solution</strong></p> <p>$ \left[\begin{array}{ccc} 2 & -4 & 1 \\ 1 & -3 & 1 \\ 3 & -7 & 2 \end{array}\right]$ $\left[\begin{array}{c} x \\ y \\ z \end{array}\right]$ = $\left[\begin{array}{c}3 \\5 \\ 12 \end{array}\right] $</p> <p>$ \left[\begin{array}{ccc|c} 2 & -4 & 1 & 3 \\ 1 & -3 & 1 & 5 \\ 3 & -7 & 2 & 12 \end{array}\right]$ - $\text { Augmented matrix. }$</p> </div> <div style='float: right; width:01%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-05-linear-inequalities-l-5-6-a6fxrbgnlbo-2.jpg"></p>

##### <Success>Solution</Success> <div style='float: left; width:99%;font-size:20px;text-align:left'> <p>$ R_1 \rightarrow \frac{1}{2} R_1$ $\left[\begin{array}{ccc|c} 1 & -2 & 1 / 2 & 3 / 2 \\ 1 & -3 & 1 & 5 \\ 3 & -7 & 2 & 12 \end{array}\right]$</p> <p>$R_2 \rightarrow R_2-R_1, \ $ $ \ R_3 \rightarrow R_3-3 R_1$</p> <p>$\left[\begin{array}{ccc|c} 1 & -2 & 1 / 2 & 3 / 2 \\ 0 & -1 & 1 / 2 & 7 / 2 \\ 0 & -1 & 1 / 2 & 15 / 2 \end{array}\right]$</p> <p>$ R_2 \rightarrow-R_2$</p> <p>$\left[\begin{array}{ccc|c} 1 & -2 & 1 / 2 & 3 / 2 \\ 0 & 1 & -1 / 2 & -7 / 2 \\ 0 & -1 & 1 / 2 & 15 / 2 \end{array}\right] $</p> <p>$R_1 \rightarrow R_1+2 R_2, \ $ $\ R_3 \rightarrow R_3+R_2$</p> <p>$\left[\begin{array}{ccc|c} 1 & 0 & -1 / 2 & -11 / 2 \\ 0 & 1 & -1 / 2 & -7 / 2 \\0 & 0 & 0 & 8 / 2 \end{array}\right]$</p> </div> <div style='float: right; width:01%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-05-linear-inequalities-l-5-6-a6fxrbgnlbo-3.jpg"></p>

### <Success>Solution</Success> <div style='float: left; width:99%;font-size:26px;text-align:left'> <p>$x-\frac{z}{2} = \frac{-11}{2}$</p> <p>$y-\frac{z}{2} = \frac{-7}{2}$</p> <p>$0.x + 0.y + 0.z = 4$</p> <p>This systems has got no solution</p> <p>$\therefore$ The given systems has no solution.</p> <p>$\left[\begin{array}{ccc|c} 1 & 0 & -1 / 2 & -11 / 2 \\ 0 & 1 & -1 / 2 & -7 / 2 \\ 0 & 0 & 0 & 8 / 2 \end{array}\right]$</p> <p>Rank $(A)=2 \ $ Rank $(\frac{A}{B})=3$</p> <p>$\Rightarrow$ The systems has no solution.</p> </div> <div style='float: right; width:01%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-05-linear-inequalities-l-5-6-a6fxrbgnlbo-4.jpg"></p>

### <Success>Example 2</Success> <div style='float: left; width:99%;font-size:26px;text-align:left'> <p>Solve the system</p> <p>$(1+i)z_1-z_2=i$</p> <p>$(1-i)z_1+(1+i)z_2=1$</p> <p><strong>Solution</strong> $\quad {\left[\begin{array}{cc|c} 1+i & -1 & i \\ 1-i & 1+i & 1 \end{array}\right]}$</p> <p>$R_1 \rightarrow \frac{1}{1+i} R_1$</p> <p>$\left[\begin{array}{cc|c} 1 & -1 / 1+i & i / 1+i \\ 1-i & 1+i & 1 \end{array}\right]$</p> <p>$=\left[\begin{array}{cc|c} 1 & \frac{-1+i}{2} & \frac{-1+i}{2} \\ 1-i & 1+i & 1 \end{array}\right]$</p> </div> <div style='float: right; width:01%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-05-linear-inequalities-l-5-6-a6fxrbgnlbo-5.jpg"> <img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-05-linear-inequalities-l-5-6-a6fxrbgnlbo-5a.jpg"></p> <p>======</p> <h3 id="successsolutionsuccess"><Success>Solution</Success></h3> <div style='float: left; width:99%;font-size:30px;text-align:left'> <p>$R_2 \rightarrow R_2-(1-i) R_1$<br> $\left[\begin{array}{cc|c} 1 & -\frac{1+i}{2} & \frac{1+i}{2} \\ 0 & 1 & 0 \end{array}\right]$</p> <p>$1+i+\frac{(1-i)^2}{2} = 1+i+ \frac{1-2i-1}{2} = 1$</p> <p>$ R_1 \rightarrow R_1-\frac{-1+i}{2} R_2$</p> <p>$\left[\begin{array}{cc|c} 1 & 0 & \frac{1+i}{2} \\ 0 & 1 & 0 \end{array}\right]$</p> <p>$z_1= \frac{1+i}{2}$ & $z_2=0$</p> <p>($\frac{1+i}{2},0$) is the solution.</p> </div>

### <Success>Example 3</Success> <div style='float: left; width:99%;font-size:28px;text-align:left'> <p>$2 x+5 y+2 z=-1 $</p> <p>$x+2 y-3 z=5 $</p> <p>$5 x+12y+z=10 $</p> <p><strong>Solution</strong></p> <p>$\left[\begin{array}{ccc} 2 & 5 & 2 \\ 1 & 2 & -3 \\ 5 & 12 & 1\end{array}\right]$ $\left[\begin{array}{c} x \\ y \\ z \end{array}\right]$ = $\left[\begin{array}{c} -1 \\ 5 \\ 10 \end{array}\right] $</p> <p>$\left[\begin{array}{ccc|c} 2 & 5 & 2 & -1 \\ 1 & 2 & -3 & 5 \\ 5 & 12 & 1 & 10 \end{array}\right] $</p> </div> <p>======</p> <h3 id="successsolutionsuccess-1"><Success>Solution</Success></h3> <div style='float: left; width:99%;font-size:25px;text-align:left'> <p>$ R_1 \rightarrow \frac{1}{2} R_1$</p> <p>$\left[\begin{array}{ccc|c} 1 & 5 / 2 & 1 & -1 / 2 \\ 1 & 2 & -3 & 5 \\ 5 & 12 & 1 & 10 \end{array}\right]$</p> <p>$ R_2 \rightarrow R_2-R_1, \ $ $ \ R_3 \rightarrow R_3-5 R_1$</p> <p>$\left[\begin{array}{ccc|c} 1 & 5 / 2 & 1 & -1 / 2 \\ 0 & -1 / 2 & -4 & 11 / 2 \\ 0 & -1 / 2 & -4 & 25 / 2 \end{array}\right] $</p> <p>$ R_2 \rightarrow -2R_2$</p> <p>$\left[\begin{array}{ccc|c} 1 & 5 / 2 & 1 & -1 / 2 \\ 0 & 1 & 8 & -11 \\ 0 & -1 / 2 & -4 & 25 / 2 \end{array}\right]$</p> </div> <div style='float: right; width:01%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-05-linear-inequalities-l-5-6-a6fxrbgnlbo-6.jpg"> <img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-05-linear-inequalities-l-5-6-a6fxrbgnlbo-6a.jpg"></p>

### <Success>Solution</Success> <div style='float: left; width:99%;font-size:30px;text-align:left'> <p>$ R_1 \rightarrow R_1-\frac{5}{2} R_2, \ \ $ $ \ R_3 \rightarrow R_2+\frac{1}{2} R_2 $</p> <p>$\left[\begin{array}{ccc|c} 1 & 0 & -19 & -54 / 2 \\ 0 & 1 & 8 & -11 \\ 0 & 0 & 0 & 14 / 2 \end{array}\right]$</p> <p>Rank (A) = 2, Rank ($\frac{A}{B}$) = 3</p> <p>$\therefore Rank(\frac{A}{B}) > Rank(A)$</p> <p>$\Rightarrow$ The given system has no solution.</p> </div> <div style='float: right; width:01%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-05-linear-inequalities-l-5-6-a6fxrbgnlbo-7.jpg">

### <Success>Example 4</Success> <div style='float: left; width:99%;font-size:30px;text-align:left'> <p>$x+3y+4z=11$</p> <p>$2x+3y+2z=7$</p> <p>$4x+9y+10z=20$</p> <p>$3x-2y+z=1$</p> </div> ====== ### <Success>Solution</Success> <div style='float: left; width:99%;font-size:30px;text-align:left'> <p>This is an over determined system.</p> <p>The number of unknowns = 3 and</p> <p>The number of equations = 4</p> <p>$\left[\begin{array}{ccc} 1 & 3 & 4 \\ 2 & 3 & 2 \\ 4 & 9 & 10 \\ 3 & -2 & 1 \end{array}\right]$ $\left[\begin{array}{c} x \\ y \\ z \end{array}\right]$ $=\left[\begin{array}{c} 11 \\ 7 \\ 20 \\ 1 \end{array}\right]$</p> </div> <p>======</p> <h3 id="successsolutionsuccess-2"><Success>Solution</Success></h3> <div style='float: left; width:99%;font-size:23px;text-align:left'> <p>$ \left[\begin{array}{ccc|c} 1 & 3 & 4 & 11 \\ 2 & 3 & 2 & 7 \\ 4 & 9 & 10 & 20 \\ 3 & -2 & 1 & 1\end{array}\right]$</p> <p>$R_2 \rightarrow R_2-2 R_1, \ \ $ $R_3 \rightarrow R_3-4 R_1, \ \ $ $R_4 \rightarrow R_4-3 R_1 $</p> <p>$\left[\begin{array}{ccc|c} 1 & 3 & 4 & 11 \\ 0 & -3 & -6 & -15 \\ 0 & -3 & -6 & -24 \\ 0 & -11 &-11 & -32 \end{array}\right] $</p> <p>$ R_2 \rightarrow \frac{1}{-3} R_2$</p> <p>$\left[\begin{array}{ccc|c} 1 & 3 & 4 & 11 \\ 0 & 1 & 2 & 5 \\ 0 & -3 & -6 & -24 \\ 0 & -11 & -11 &-32 \end{array}\right] $</p> </div> <div style='float: right; width:01%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-05-linear-inequalities-l-5-6-a6fxrbgnlbo-8.jpg"> <img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-05-linear-inequalities-l-5-6-a6fxrbgnlbo-8a.jpg"></p>

### <Success>Solution</Success> <div style='float: left; width:99%;font-size:27px;text-align:left'> <p>$ R_1 \rightarrow R_1-3 R_2, \ \ $ $ R_3 \rightarrow R_3+3 R_2, \ \ $ $ R_4 \rightarrow R_4+11 R_2$</p> <p>$\left[\begin{array}{ccc|c} 1 & 0 & -2 & -4 \\ 0 & 1 & 2 & 5 \\ 0 & 0 & 0 & -9 \\ 0 & 0 & 11 & 3\end{array}\right]$</p> <p>$ R_3 \longleftrightarrow R_4$</p> <p>$\left[\begin{array}{ccc|c} 1 & 0 & -2 & -4 \\ 0 & 1 & 2 & 5 \\ 0 & 0 & 11 & 3 \\ 0 & 0 & 0 & -9\end{array}\right]$</p> <p>$\therefore$ The systems has got no solution.</p> </div> <div style='float: right; width:01%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-05-linear-inequalities-l-5-6-a6fxrbgnlbo-9.jpg"></p>

### <Success>Example 5</Success> <div style='float: left; width:99%;font-size:27px;text-align:left'> <p>$x+2y+3t=7$</p> <p>$z+4t=8$</p> <p><strong>Solution</strong></p> <p>This is an under determined systems.</p> <p>Number of unknowns = 4 and number of equations = 2</p> <p>$\left[\begin{array}{cccc} 1 & 2 & 0 & 3 \\ 0 & 0 & 1 & 4 \end{array}\right]$ $\left[\begin{array}{c}x \\ y \\ z \\ t \end{array}\right]$=$\left[\begin{array}{c} 7 \\ 8 \end{array}\right]$</p> <p>Variables $x$ & $z$ are dependent variables. Variables $y$ & $t$ are independent variables.</p> </div> <div style='float: right; width:01%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-05-linear-inequalities-l-5-6-a6fxrbgnlbo-10.jpg"></p>

### <Success>Solution</Success> <div style='float: left; width:99%;font-size:27px;text-align:left'> <p>$y=\lambda$ & $t=\mu \ $ where $\lambda$ & $\mu$ are any arbitrary real numbers.</p> <p>$x+2 y+3 t=7 $</p> <p>$z+4 t=8 $</p> <p>$x+2 \lambda+3 \mu=7 $</p> <p>$z+4 \mu=8 $</p> <p>$\Rightarrow z=8-4 \mu$</p> <p>& $x=7-2 \lambda-3 \mu$</p> <p>The solution set to this systems is $\biggl \{(7-2\lambda-3\mu, \lambda, 8-4\mu, \mu) | \lambda, \mu \in \mathbb{R} \biggl \}.$</p> </div> <div style='float: right; width:01%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-05-linear-inequalities-l-5-6-a6fxrbgnlbo-11.jpg"> <img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-05-linear-inequalities-l-5-6-a6fxrbgnlbo-11a.jpg"></p>

### <Success>Example 6</Success> <div style='float: left; width:99%;font-size:30px;text-align:left'> <p>$8 x+5 y+11 z=30 $</p> <p>$ -x-4 y+2 z=3 $</p> <p>$ 2 x-y+5 z=12$</p> <p><strong>Solution</strong></p> <p>$ \left[\begin{array}{ccc} 8 & 5 & 11 \\ -1 & -4 & 2 \\ 2 & -1 & 5\end{array}\right]$ $\left[\begin{array}{l} x \\ y \\ z \end{array}\right]$=$\left[\begin{array}{c} 30 \\ 3 \\ 12\end{array}\right]$</p> <p>$ {\left[\begin{array}{ccc|c} 8 & 5 & 11 & 30 \\ -1 & -4 & 2 & 3 \\ 2 & -1 & 5 & 12 \end{array}\right]}$</p> </div> <div style='float: right; width:01%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-05-linear-inequalities-l-5-6-a6fxrbgnlbo-12.jpg"></p>

### <Success>Solution</Success> <div style='float: left; width:99%;font-size:25px;text-align:left'> <p>$ R_1 \rightarrow \frac{1}{8} R_1$</p> <p>$\left[\begin{array}{ccc|c} 1 & 5 / 8 & 11 / 8 & 30 / 8 \\ -1 & -4 & 2 & 3 \\ 2 & -1 & 5 &12\end{array}\right]$</p> <p>$ R_2 \rightarrow R_2+R_1, \ $ $\ R_3 \rightarrow R_3-2 R_1 $</p> <p>$\left[\begin{array}{ccc|c} 1 & 5 / 8 & 11 / 8 & 30 / 8 \\ 0 & -27 / 8 & 27 / 8 & 54 / 8 \\ 0 & 1 / 4 & 9 / 4 & 18 / 4 \end{array}\right]$</p> <p>$ R_2 \rightarrow-\frac{-8}{27} R_2$</p> <p>$\left[\begin{array}{ccc|c} 1 & 5 / 8 & 11 / 8 & 30 / 8 \\ 0 & 1 & -1 & -2 \\ 0 & 1 / 4 & 9 / 4 & 18/ 4 \end{array}\right]$</p> </div> <div style='float: right; width:01%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-05-linear-inequalities-l-5-6-a6fxrbgnlbo-14.jpg"></p>

### <Success>Solution</Success> <div style='float: left; width:99%;font-size:20px;text-align:left'> <p>$R_1 \rightarrow R_1-\frac{5}{8} R_2, \ $ $\ R_3 \rightarrow R_3-\frac{1}{4} R_2 $</p> <p>$\left[\begin{array}{ccc|c} 1 & 0 & 16 / 8 & 40 / 8 \\ 0 & 1 & -1 & -2 \\ 0 & 0 & 10 / 4 & 20 / 4\end{array}\right] $</p> <p>$ R_3 \rightarrow \frac{4}{10} R_3$</p> <p>$\left[\begin{array}{ccc|c} 1 & 0 & 16 / 8 & 40 / 8 \\ 0 & 1 & -1 & -2 \\ 0 & 0 & 1 & 2\end{array}\right] $</p> <p>$R_2 \rightarrow R_2+R_3, \ \ $ $\ R_1 \rightarrow R_1-\frac{16}{8} R_3$</p> <p>$\left[\begin{array}{ccc|c} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 2 \end{array}\right]$</p> <p>$(1,0,2)$ is the solution.</p> </div> <div style='float: right; width:01%;'> <p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-05-linear-inequalities-l-5-6-a6fxrbgnlbo-15.jpg"> <img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-05-linear-inequalities-l-5-6-a6fxrbgnlbo-15a.jpg"></p> </div> <p>======</p> <h3 id="successthank-yousuccess"><Success>Thank you</Success></h3> <div style='float: left; width:99%;font-size:20px;text-align:left'> </div>