Question

determine if the following set of equations has infinitely many solutions or no solutions

x-y+2z+w = 4

-2x+3y=-4

x+y+z-4w=3

Answer 1

Noco. The given System is> x-ry ry +27 + W = 4 -2x + 3y = (-4). x + y +2 -4W = 3. .Now the Coeffient Matrix and Aug men ted Matrix of the system are -1 2 1 1 = 1 1 2 1 4 -2 3 0 O -4 1 1 1-4 3 & A = 3 o ܝܢ ܠܐ -2 3 0 1 1-4 Now, Let us apply elementary Row operation on Ā to Reduce it in Row echelon Form. Rø=Rx+28 Riartre 4 2 4 R = R3-R R₂ =R3 -2R2 1 1 2 1 4 o 1 > 0 2 -1 -5 1 6 3 8 8 1 6 o 0 3 ܥܝܐ ( 4 R 0 1 2 4 4 4 2 4 0 1 0 -g-g 1 O 0 -9 1 1 0 o o -3 2 1 0 1 0 -2 0 0 1 1 1 O R=R-6R3 R2=R2-4R2 tem Here R(A) R(A) =3. is consistent. Now, This system is equivalent y-2W = 0 , 7+W = 1. The sys and. Then to 2-3W = 2 9

Now, Let w=k Where KER. Then 2 = (2+3k) y = 2K 3 7 = I-K. Then, the Solution is (2.9. 2, w) (2+3K, 2K, 1 1-K, K) So, the system has Infinitely Many Solution. where KER