Use matrices with Gaussian elimination to solve the system of equations.

Question

Use matrices with Gaussian elimination to solve the system of equations. (If there is no solution, enter NO SOLUTION. If there are infinitely many solutions, enter INFINITELY MANY.)

 

x + z = −4
y − z = 1
x + y = −3

Answer 1

We write the matrix for the system.

x   y   z   b

1   0   1  -4            row R₁

0   1   -1  1            row R₂

1   1    0  -3           row R₃

 

When I write R₁ - R₃

I mean I am replacing the indicated row with the difference of row 1 minus row 3.

 

1 0 1 -4
0 1 -1 1 
1 1 0 -3      R₁-R₃

 

1   0  1  -4
0   1 -1   1
0  -1  1  -1     R₂+R₃

 

1 0 1 -4
0 1 -1 1
0 0  0 0

 

This system has infinitely many solutions.

The independent variables are x and y.