Determine whether the system has a unique solution, infinitely many solutions, or no solution

  To determine the nature of the solution for the given system of linear equations: 2x + 3y - z = 5 …(1) x - y + 2z = -3 …(2) 3x + 2y + 4z = 10 …(3) We can use the method of Gaussian elimination or matrix operations to simplify the system and identify the solution. First, we can represent the system of equations in matrix form as: | 2 3 -1 | | x | | 5 | | 1 -1 2 | * | y | = | -3 | | 3 2 4 | | z | | 10 | By performing row operations to simplify the augmented matrix, we can determine the nature of the solution. Performing the row operation R2 = R2 - (1/2)R1 and R3 = R3 - (3/2)R1, we get: | 2 3 -1 | | x | | 5 | | 0 -2.5 2.5| * | y | = | -5 | | 0 -3.5 5 | | z | | -2.5 | Next, we can perform the row operation R3 = R3 - (7/5)R2: | 2 3 -1 | | x | | 5 | | 0 -2.5 2.5| * | y | = | -5 | | 0 0 -1 | | z | | -1 | Now, let’s solve for z using the third equation: z = -1 Substituting this value back into the second equation, we can solve for y: -2.5y + 2.5(-1) = -5 -2.5y - 2.5 = -5 -2.5y = -2.5 y = 1 Finally, substituting the values of y and z into the first equation, we can solve for x: 2x + 3(1) - (-1) = 5 2x + 3 + 1 = 5 2x + 4 = 5 2x = 1 x = 0.5 Therefore, the solution to the system of linear equations is: x = 0.5, y = 1, z = -1 Hence, the system has a unique solution.  

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