Question

#5 Determine for which values of the system 3x + 2y x + y +z = 2x + 2y + Az has: 4 p. a) one solution; 4 p. b) no solution. 4 p. c) infinitely many solutions;

Answer 1

systemAx=b is

7320) 111 12 2 )

augmented matrix is

320 1 1112 (2230

R2 + R- RI

3 2 0 1 0 1 2 20 u

R3 + R3- RI

coll001N COIN

R2 + R

تا تانتانت و اهانت

Rz + R3- R

27 ܝ2 0 0| ܘ ELܒ

$R_3 \leftarrow R_3\cdot 2$

$=\begin{pmatrix}3&2&0&1\\ 0&\frac{2}{3}&\:\lambda \:&-\frac{2}{3}\\ 0&0&2-\:\lambda \:&4\end{pmatrix}$

when  $2-\lambda \neq 0$ then the system has one solution

$2-\lambda \neq 0$

${\color{Red} \lambda \neq 2}$....................for one solution

.

.

.

when   then the system has no solution

2 - X=0

2 - X=0

${\color{Red} \lambda =2}$....................for no solution

.

.

.

there is no value of $\lambda$ that makes the system has infinitely many solutions

${\color{Red} infinitely\:\:many\:\:solutions:Not\:Possible\:}$

Answer 2

systemAx=b is

7320) 111 12 2 )

augmented matrix is

320 1 1112 (2230

R2 + R- RI

3 2 0 1 0 1 2 20 u

R3 + R3- RI

coll001N COIN

R2 + R

تا تانتانت و اهانت

Rz + R3- R

27 ܝ2 0 0| ܘ ELܒ

$R_3 \leftarrow R_3\cdot 2$

$=\begin{pmatrix}3&2&0&1\\ 0&\frac{2}{3}&\:\lambda \:&-\frac{2}{3}\\ 0&0&2-\:\lambda \:&4\end{pmatrix}$

when  $2-\lambda \neq 0$ then the system has one solution

$2-\lambda \neq 0$

${\color{Red} \lambda \neq 2}$....................for one solution

.

.

.

when   then the system has no solution

2 - X=0

2 - X=0

${\color{Red} \lambda =2}$....................for no solution

.

.

.

there is no value of $\lambda$ that makes the system has infinitely many solutions

${\color{Red} infinitely\:\:many\:\:solutions:Not\:Possible\:}$