3x + y = -4 -6x-2y= 3 Solve by using the Gauss-Jordan method. 2x - y = 5 x + 2y = -5 Solution: -1 51 2 R₁ = R₂ 2 -2R1+ R₂⇒ R₂ 15 1 2 -R2⇒ R₂ 10 -2R₂+ R₁⇒ R₁ = C = [] Set up the augmented matrix. Switch row 1 and row 2 to get a 1 in the upper left position. Multiply row 1 by -2 and add the result to row 2. This produces an entry of 0 below the upper left position. Multiply row 2 by - to produce a 1 along the diagonal in the second row. Multiply row 2 by -2 and add the result to row 1. This produces a0 in the first row, second column. The matrix C is in reduced row echelon form. From the augmented matrix, we have x= 1 and y=-3. The solution set is {(1, -3)}. Solve by using the Gauss-Jordan method. x -2x = −y+5 +2zy-10 3x+6y+7z14 Solution: First write each equation in the system in standard form. x =-y+5- -2x 2z y 10 3x+6y+7z 14 - 2R1 R2 R2- -3R₁+R3 R3- -IR₂+R₁⇒R₁ -3R2+R3 R3- + + x + y = = 5 -2xy + 2z = -10 3x+6y+7z 14 1 1 0 51 Set up the -2-1 2 -10 augmented matrix. 3 6 7 14] [1 1 0 0 1 2 0 LO 3 7 [10-2 51 0 1 2 LO O Multiply row 1 by 2 and add the result to row 2. Multiply row 1 by -3 and add the result to row 3. Multiply row 2 by -1 and add the result to row 1. Multiply row 2 by -3 and add the result to row 3. 2R3 + R₁ ⇒ R₁ -2R3+ R₂R2. + [1 0 0 31 Lo 0 01 0 1 2 Multiply row 3 by 2 and add the result to row 1. Multiply row 3 by -2 and add the result to row 2. From the reduced row echelon form of the matrix, we have x = 3, y = 2, and z = -1. The solution set is {(3, 2, -1)}. Solve by using the Gauss-Jordan method. x- - 3y = 4 1 3 2 – y = 2 --2 Solution: -3 -R₁ + R₂⇒ R₂ - | 21 [1-3 41 Set up the augmented matrix. [3] Multiply row 1 by — and add the 0 result to row 2. The second row of the augmented matrix represents the equation 0 = 0. The equations are dependent. The solution set is {(x, y) lx-3y= 4}. Solve by using the Gauss-Jordan method. x+3y=2 -3x-9y=1 Solution: 3 3R₁ + R₂⇒ R₂ 3 Set up the augmented matrix. Multiply row 1 by 3 and add the result to row 2. The second row of the augmented matrix represents the contradiction 0=7. The system is inconsistent. There is no solution,{ }.