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

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

Use a software program or a graphing utility to solve the system
of linear equations. (Round your values to three decimal places. If
there is no solution, enter NO SOLUTION. If the system has an
infinite number of solutions, set
x4 = t
and solve for x1, x2,
and x3 in terms of t.)
x1
−
2x2
+
5x3
−
3x4
=
23.3
x1
+
4x2
−
7x3
−
2x4
=
45.4
3x1
−
5x2
+
7x3
+
4x4
=...

Use a software program or a graphing utility to solve the system
of linear equations. (Round your values to three decimal places. If
there is no solution, enter NO SOLUTION. If the system has an
infinite number of solutions, set
x4 = t
and solve for x1, x2,
and x3 in terms of t.)
x1
−
2x2
+
5x3
−
3x4
=
23.3
x1
+
4x2
−
7x3
−
2x4
=
45.4
3x1
−
5x2
+
7x3
+
4x4
=...

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