Systems of Linear Equations Questions

Question

Find all solutions to the following systems of linear equations:

(a) x + y = −5

2x − 4y = −4

(b) x + y − z = 1

x + 2y + 2z = 1

2x + y − 5z = 2

Answer 1

Part (a.) has a finite solution, while Part (b.) was found to have infinite solutions on one variable. The solutions to each is imaged below.

Part (a.): Set up the matrix as:

1.) Multiply 1^st  row by (-2) and add it to 2nd row

1 1 -5

2 -4 -4

2.) Divide 2^nd row by (6)

-2 -2 10

0 -6 6

3.) Add 2^nd row to 1^st row and multiply 2^nd row by (-1)

1 1 -5

0 -1 1

4.) Thus x = -4 and y = -1

1 0 -4

0 1 -1

Part (b.) is solved as below:

1.) Multiply 1^st row by (-1) and add that to 2^nd row.

          2.) Multiply 1^st row by (-2) and add that to 3^rd row.

1 1 -1 1

1 2 2 1       

2 1 -5 2

  3.) Add 2^nd row to the 3^rd row.

1 1 -1 1

0 1 3 0

0 -1 -3 0

4.) Since the 3^rd row is all zero’s, this translates to the “z” variable to be any value.

1 1 -1 1

0 1 3 0

0 0 0 0

Thus, these equations have infinite solutions for its “z” value.