Systems of Equations/Matrices

Question

Solve each of the following systems of equations. If the system has a unique solution (one solution), state this and exhibit the solution. If the system is inconsistent (has no solutions) state this. If the system has an infinite number of solutions, state this and exhibit the parametric solution

x + y + z + w = 6

2x + 3y - 2z + 5w = 7

3x - 2y + 5z - w = 7

x + 5y - 4z + 2w = 8

4x - 3y + 7z - 2w = 9

[1 0 0 0 1] [1 0 0 0 1]

[0 1 0 0 3] [0 1 0 0 3]

[0 0 1 0 2] is this row equivalent to [0 0 1 0 2]

[0 0 0 1 0] [0 0 0 1 0]

[0 0 0 0 0] [0 0 0 0 0]

Answer 1

the solution is {x=1, y=3, z=2, w=0}

1 1 1 1 6

2 3 -2 5 7

3 -2 5 -1 7

1 5 -4 2 8

4 -3 7 -2 9

===========================

-2*row1 + row2;

-3*row1 + row3;

-row1+row4

-4*row1 + row5

1 1 1 1 6

0 1 -4 3 -5

0 -5 2 -4 -11

0 4 -5 1 2

0 -7 3 -6 -15

=====================================

5 * row2 + row3

-4* row2+row4

7 * row2 + row5

1 1 1 1 6

0 1 -4 3 -5

0 0 -18 11 -36

0 0 11 -11 22

0 0 -25 15 -50

==========================================

r4/11; r5/-5

1 1 1 1 6

0 1 -4 3 -5

0 0 -18 11 -36

0 0 1 -1 2

0 0 5 -3 10

===============================================

-5*row4 + row5

1 1 1 1 6

0 1 -4 3 -5

0 0 -18 11 -36

0 0 1 -1 2

0 0 0 2 0

backwards substitution phase:

so from the last row, the equation is 2w=0 ---> w = 0

From row 4, the equation is z-w = 2;

z-0 = 2

z =2

From row 2, the equation is y-4z + 3w = -5

y - 4(2) + 3(0) = -5

y - 8 = -5

y = 3

Finally the first row says the total of the vars is 6.

x+ 3+2+0 = 6

x =1

the solution is {x=1, y=3, z=2, w=0}