Q:

Solve the following system of linear equations: 3x1+6x2+6x3 = -9 -2x1โ€“3x2-3x3 = 3 If the system has infinitely many solutions, your answer may use expressions involving the parameters r, s, and t. O The system has at least one solution x1 = 0 x2 = 0 X3 = 0 O O

Accepted Solution

A:
Answer:The set of solutions is [tex]\{\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}12\\-7-r\\r\end{array}\right]: \text{r is a real number} ย \}[/tex]Step-by-step explanation:The augmented matrix of the system is [tex]\left[\begin{array}{ccccc}3&6&6&-9\\-2&-3&-3&3\end{array}\right][/tex].We will use rows operations for find the echelon form of the matrix.In row 2 we subtract [tex]\frac{2}{3}[/tex] from row 1. (R2- 2/3R1) and we obtain the matrix [tex]\left[\begin{array}{cccc}3&6&6&-9\\0&1&1&-7\end{array}\right][/tex]We multiply the row 1 by [tex]\frac{1}{3}[/tex].Now we solve for the unknown variables:[tex]x_2+x_3=-7[/tex], [tex]x_2=-7-x_3[/tex][tex]x_1+2x_2+2x_3=-2[/tex], [tex]x_1+2(-7-x_3)+2x_3=-2[/tex] then [tex]x_1=12[/tex]The system has a free variable, the the system has infinite solutions and the set of solutions is [tex]\{\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}12\\-7-r\\r\end{array}\right]: \text{r is a real number} ย \}[/tex]