Question

3. Consider the system of linear equations

3x^{1} + x^{2} + 4x^{3} − x^{4}
= 7

2x^{1} − 2x^{2} − x^{3} + 2x^{4}
= 1

5x^{1} + 7x^{2} + 14x^{3} −
8x^{4} = 20

x^{1} + 3x^{2} + 2x^{3} + 4x^{4}
= −4

b) Solve this linear system applying Gaussian forward elimination with partial pivoting and back ward substitution, by hand. In (b) use fractions throughout your calculations.

(i think x^{1} = 1, x^{2}= -1, x^{3} =1,
x^{4}=-1, but i am not sure)

Answer #1

solve the following linear system by gauss-jordan
method
x1 + x2 - 2x3 + x4 = 8
3x1 - 2x2 - x4 = 3
-x1 + x2 - x3 + x4 = 2
2x1 - x2 + x3 - 2x4 = -3

in parts a and b use gaussian elimination to solve the systems
of linear equations. show all steps.
a. x1 - 4x2 - x3 + x4 = 3
3x1 - 12 x2 - 3x4 = 12
2x1 - 8x2 + 4x3 - 10x4 = 12
b. x1 + x2 + x3 - x4 = 2
2x1 + 2x2 - 2x3 = 3
2x1 + 2x2 - x4 = 2

Linear Algebra
find all the solutions of the linear system using Gaussian
Elimination
x1-x2+3x3+2x4=1
-x1+x2-2x3+x4=-2
2x1-2x2+7x3+7x4=1

Solve for all 4-tuples (x1, x2, x3, x4) simultaneously
satisfying the following equations:
8x1 −9x2 −2x3 −5x4 = 100
9x1 +6x2 −6x3 +9x4 = 60
−3x1 −9x2 +4x3 −2x4 = −52
−7x2 +8x3 +8x4 = −135

Use Gauss
Elimination with partial pivoting method to find x1, x2,and x3 for
the following set of linear equations. You should show all your
work in details. Verify your solutions
2X1
+ X2 - X3 = 1
5X1
+ 2X2 + 2X3 = -4
3X1
+ X2 + X3 = 5

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 linear systems that abides by the following rules.
Show all steps.
I. The first nonzero coefficient in each equation is one.
II. If an unknown is the first unknown with a nonzero
coefficient in some equation, then that unknown doesn't appear in
other equations.
II. The first unknown to appear in any equation has a larger
subscript than the first unknown in any preceding equation.
a. x1 + 2x2 - 3x3 + x4 = 1.
-x1 - x2...

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

Duality Theory: Consider the following LP:
max 2x1+2x2+4x3
x1−2x2+2x3≤−1
3x1−2x2+4x3≤−3
x1,x2,x3≤0
Formulate a dual of this linear program. Select all the correct
objective function and constraints
1. min −y1−3y2
2. min −y1−3y2
3. y1+3y2≤2
4. −2y1−2y2≤2
5. 2y1+4y2≤4
6. y1,y2≤0

Use Gaussian elimination to solve the following system of linear
equations.
2x1 -2x2 -x3
+6x4 -2x5=1
x1 - x2
+x3 +2x4 - x5=
2
4x1 -4x2
-5x3 +7x4
-x5=6

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