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Determine the values of a and b for which the system {[2x+y+az=-1],[3x-2y+z=b],[5x-8y+9z=3]:} (a) has no solution, (b) has only one solution, (c) has infinitely many solutions.

Determine the values of a a and b b for which the system {2x+y+az=−13x−2y+z=b5x−8y+9z=3 \left\{\begin{array}{l}2 x+y+a z=-1 \\ 3 x-2 y+z=b \\ 5 x-8 y+9 z=3\end{array}\right. \newline(a) has no solution, (b) has only one solution, (c) has infinitely many solutions.

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Q. Determine the values of a a and b b for which the system {2x+y+az=−13x−2y+z=b5x−8y+9z=3 \left\{\begin{array}{l}2 x+y+a z=-1 \\ 3 x-2 y+z=b \\ 5 x-8 y+9 z=3\end{array}\right. \newline(a) has no solution, (b) has only one solution, (c) has infinitely many solutions.
  1. Analyze Coefficient Matrix: Analyze the coefficient matrix of the system to determine the conditions for different types of solutions. The system of equations is:\newline2x+y+az=−13x−2y+z=b5x−8y+9z=3 \begin{align*} 2x + y + az &= -1 \\ 3x - 2y + z &= b \\ 5x - 8y + 9z &= 3 \end{align*}
  2. Write Augmented Matrix: Write the augmented matrix and perform row reduction:\newline[21a∣−13−21∣b5−89∣3] \begin{bmatrix} 2 & 1 & a & | & -1 \\ 3 & -2 & 1 & | & b \\ 5 & -8 & 9 & | & 3 \end{bmatrix}
  3. Perform Row Reduction: Use elementary row operations to simplify the matrix. Start by making the first element of the first row a 11 by dividing the entire first row by 22:\newline[10.5a/2∣−0.53−21∣b5−89∣3] \begin{bmatrix} 1 & 0.5 & a/2 & | & -0.5 \\ 3 & -2 & 1 & | & b \\ 5 & -8 & 9 & | & 3 \end{bmatrix}
  4. Simplify Matrix: Subtract 33 times the first row from the second row and 55 times the first row from the third row:\newline[10.5a/2∣−0.50−3.51−1.5a∣b+1.50−10.59−2.5a∣5.5] \begin{bmatrix} 1 & 0.5 & a/2 & | & -0.5 \\ 0 & -3.5 & 1 - 1.5a & | & b + 1.5 \\ 0 & -10.5 & 9 - 2.5a & | & 5.5 \end{bmatrix}
  5. Make Leading Coefficient 11: Divide the second row by −3-3.55 to make the leading coefficient of the second row a 11:\newline[10.5a/2∣−0.5011−1.5a−3.5∣b+1.5−3.50−10.59−2.5a∣5.5] \begin{bmatrix} 1 & 0.5 & a/2 & | & -0.5 \\ 0 & 1 & \frac{1 - 1.5a}{-3.5} & | & \frac{b + 1.5}{-3.5} \\ 0 & -10.5 & 9 - 2.5a & | & 5.5 \end{bmatrix}
  6. Eliminate Y-Term: Add 1010.55 times the second row to the third row to eliminate the y-term in the third row:\newline[10.5a/2∣−0.5011−1.5a−3.5∣b+1.5−3.5009−2.5a+10.5⋅1−1.5a−3.5∣5.5+10.5⋅b+1.5−3.5] \begin{bmatrix} 1 & 0.5 & a/2 & | & -0.5 \\ 0 & 1 & \frac{1 - 1.5a}{-3.5} & | & \frac{b + 1.5}{-3.5} \\ 0 & 0 & 9 - 2.5a + 10.5 \cdot \frac{1 - 1.5a}{-3.5} & | & 5.5 + 10.5 \cdot \frac{b + 1.5}{-3.5} \end{bmatrix}

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