\( \left\{ \begin{array} { l } { 2 x + 3 y + z = 1 } \\ { 6 x - 2 y - z = - 1 4 } \\ { 3 x + y - z = 1 } \end{array} 6 \left\{ \begin{array} { l } { 5 x - 2 y + z = 2 4 } \\ { 2 x + 5 y - 2 z = - 1 4 } \\ { x - 4 y + 3 z = 2 6 } \end{array} 7 \left\{ \begin{array} { l } { 4 x + 2 y + 3 z = 8 } \\ { 3 x + 4 y + 2 z = - 1 } \\ { 2 x - y + 5 z = 3 } \end{array} \right. \right. \right. \)

It seems like you've provided a series of systems of linear equations, but the formatting is a bit unclear. Let's break down the systems of equations you've provided and solve them one by one. ### System 1: \[ \begin{cases} 2x + 3y + z = 1 \quad (1) \\ 6x - 2y - z = -14 \quad (2) \\ 3x + y - z = 1 \quad (3) \end{cases} \] ### System 2: \[ \begin{cases} 5x - 2y + z = 24 \quad (4) \\ 2x + 5y - 2z = -14 \quad (5) \\ x - 4y + 3z = 26 \quad (6) \end{cases} \] ### System 3: \[ \begin{cases} 4x + 2y + 3z = 8 \quad (7) \\ 3x + 4y + 2z = -1 \quad (8) \\ 2x - y + 5z = 3 \quad (9) \end{cases} \] Let's solve each system one by one. ### Solving System 1: 1. From equation (1): \( z = 1 - 2x - 3y \) 2. Substitute \( z \) into equations (2) and (3): - For (2): \[ 6x - 2y - (1 - 2x - 3y) = -14 \implies 6x - 2y - 1 + 2x + 3y = -14 \implies 8x + y = -13 \quad (10) \] - For (3): \[ 3x + y - (1 - 2x - 3y) = 1 \implies 3x + y - 1 + 2x + 3y = 1 \implies 5x + 4y = 2 \quad (11) \] 3. Now we have a new system: \[ \begin{cases} 8x + y = -13 \quad (10) \\ 5x + 4y = 2 \quad (11) \end{cases} \] 4. Solve for \( y \) from (10): \[ y = -13 - 8x \] 5. Substitute into (11): \[ 5x + 4(-13 - 8x) = 2 \implies 5x - 52 - 32x = 2 \implies -27x = 54 \implies x = -2 \] 6. Substitute \( x = -2 \) back into (10): \[ 8(-2) + y = -13 \implies -16 + y = -13 \implies y = 3 \] 7. Substitute \( x = -2 \) and \( y = 3 \) back into (1) to find \( z \): \[ z = 1 - 2(-2) - 3(3) = 1 + 4 - 9 = -4 \] Thus, the solution for System 1 is: \[ (x, y, z) = (-2, 3, -4) \] ### Solving System 2: 1. From equation (4): \( z = 24 - 5x + 2y \) 2. Substitute \( z \) into equations (5) and (6): - For (5): \[ 2x + 5y - 2(24 - 5x + 2y) = -14 \implies 2x + 5y - 48 + 10x - 4y = -14 \implies 12x + y = 34 \quad (12) \] - For (6): \[ x - 4y + 3(24 - 5x + 2y) = 26 \implies x - 4y + 72 - 15x + 6y = 26 \implies -14x + 2y = -46 \implies 7x - y = 23 \quad (13) \] 3. Now we have a new system: \[ \begin{cases} 12x + y = 34 \quad (12) \\ 7x - y = 23 \quad (13) \end{cases} \