Systems of Equations

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 + 3z + w = 6

2x + 3y + 7z + 5w = 21

3x - 2y + 4z - w = -5

x + 5y + 7z + 2w = 20

4x - 3y + 5z - 2w = -9

[1 1 3 1 6] Row 1 Row 2 [1 0 2 0 1]

[2 3 7 5 21] [0 1 1 0 3]

[3 -2 4 -1 -5] is this row equivalent to [0 0 0 1 2}

[1 5 7 2 20] [0 0 0 0 0]

[4 -3 5 -2 -9] [0 0 0 0 0]

Answer 1

1 1 3 1 6

2 3 7 5 21

3 -2 4 -1 -5

1 5 7 2 20

4 -3 5 -2 -9

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-2*row1 + row2;

-3*row1 + row3;

-1*row1 + row4;

-4*row1 + row5;

1 1 3 1 6

0 1 1 3 9

0 -5 -5 -4 -23

0 4 4 1 14

0 -7 -7 -6 -33

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5*row2 + row3

-4*row2 + row4

7*row2 + row5

1 1 3 1 6

0 1 1 3 9

0 0 0 11 22

0 0 0 -11 -22

0 0 0 15 30

the last three rows give w = 2

the second row becomes y + z + 3w = 9

y + z + 3(2) = 9

y + z + 6 = 9

y + z = 3

y = 3-z

the first row becomes x + y + 3z + w = 6

x + y + 3z + 2 = 6

x + y + z + 2z + 2 = 6

x + (3) + 2z + 2 = 6

x + 2z + 5 = 6

x + 2z = 1

x = 1 - 2z

So there are infinitely many solutions..

In terms of free variable z, they are

{x=1-2z, y = 3-z, z, 2}

For example, if z=0, then the particular solution is {1,3,0,2}

For example, if z=1, then the particular solution is {-1,2,1,2}