Question

Solve the following system of linear equations: 3x2−9x3 = −3 x1−2x2+x3 = 2 x2−3x3 = 0 If the system has no solution, demonstrate this by giving a row-echelon form of the augmented matrix for the system. If the system has infinitely many solutions, your answer may use expressions involving the parameters r, s, and t. You can resize a matrix (when appropriate) by clicking and dragging the bottom-right corner of the matrix.

Answer #1

The given homogeneous linear equations are
3x_{2}−9x_{3}=0 or,
x_{2}−3x_{3}=0…(1),−3x_{1}−2x_{2}+x_{3}
= 0 or, 3x_{1}+2x_{2}-x_{3} = 0…(2)and
2x_{2}−3x_{3} = 0…(3).

The coefficient matrix of the given homogeneous system of linear equations is A (say) =

0 |
1 |
-3 |

3 |
2 |
-1 |

0 |
2 |
-3 |

To solve the given homogeneous system of linear equations, we will reduce A to its RREF as under:

Interchange the 1st row and the 2nd row

Multiply the 1st row by 1/3

Add -2 times the 2nd row to the 3rd row

Multiply the 3rd row by 1/3

Add 3 times the 3rd row to the 2nd row

Add 1/3 times the 3rd row to the 1st row

Add -2/3 times the 2nd row to the 1st row

Then the RREF of A is I_{3}.

Hence, the only solutionn to the given homogeneous system of linear
equations is the trivial solution i.e. x_{1}=0,
x_{2} = 0 and x_{3} = 0.

Consider the following system of linear equations:
2x1−2x2+4x3
=
−10
x1+x2−2x3
=
5
−2x1+x3
=
−2
Let A be the coefficient matrix and X the solution matrix to the
system. Solve the system by first computing A−1 and then
using it to find X.
You can resize a matrix (when appropriate) by clicking and dragging
the bottom-right corner of the matrix.

Consider the following system of equations.
x1- x2+ 3x3 =2
2x1+ x2+ 2x3 =2
-2x1 -2x2 +x3 =3
Write a matrix equation that is equivalent to the system of
linear equations.
(b) Solve the system using the inverse of the coefficient
matrix.

Give augmented matrix for this system. Find all solutions to
this system. Indicate all parameters.
x1-x2+x3+x4=1
2x2+3x3+4x4=2
x1-x2+2x3+3x4=3
x1=? x2=? x3=? x4=?

2X1-X2+X3+7X4=0
-1X1-2X2-3X3-11X4=0
-1X1+4X2+3X3+7X4=0
a. Find the reduced row - echelon form of the coefficient
matrix
b. State the solutions for variables X1,X2,X3,X4 (including
parameters s and t)
c. Find two solution vectors u and v such that the solution
space is \
a set of all linear combinations of the form su + tv.

Solve the 3x3 system.
x1-x2+x3=3
-2x1+3x2+2x3=7
3x1-3x2+2x3=6

Consider the following system of equations.
x1+2x2+2x3 −
2x4+2x5 = 5
−2x1 − 4x3+ x4 −
10x5 = −11
x1+2x2 − x3+3x5 =
4
1. Represent the system as an augmented matrix.
2. Reduce the matrix to row reduced echelon form. (This can be
accomplished by hand or by MATLAB. No need to post code.)
3. Write the set of solutions as a linear combination of vectors
in R5. (This must be accomplished by hand using the rref
form found...

Solve the following systems by forming the augmented matrix and
reducing to reduced row echelon form. In each case decide whether
the system has a unique solution, infinitely many solutions or no
solution. Show pivots in squares. Describe the solution set.
-3x1+x2-x3=10
x2+4X3=12
-3x1+2x2+3x3=11

Solve the system
-2x1+4x2+5x3=-22
-4x1+4x2-3x3=-28
4x1-4x2+3x3=30
a)the initial matrix is:
b)First, perform the Row Operation 1/-2R1->R1. The resulting
matrix is:
c)Next perform operations
+4R1+R2->R2
-4R1+R3->R3
The resulting matrix is:
d) Finish simplyfying the augmented mantrix down to reduced row
echelon form. The reduced matrix is:
e) How many solutions does the system have?
f) What are the solutions to the system?
x1 =
x2 =
x3 =

Solve the LPP below by making use of the dual simplex
method.
min z=2x1+3x2+4x3
st: x1+2x2+x3>=3
2x1-x2+3x3>=4
x1,x2,x3>=0

Consider the following
LP: Max Z=X1+5X2+3X3
s.t. X1+2X2+X3=3
2X1-X2 =4 X1,X2,X3≥0
a.) Write the associated dual model
b.) Given the information that the optimal basic variables are
X1 and X3, determine the associated optimal dual solution.

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