Determine the values of the constants a and b so that the system
 x-2y+3x=4
2x-3y+az=5
3x-4y+5z=b
has 1,no solution
       2,unique solution
       3,infinitly many solutions

Let's consider the system of linear equations provided:

\[
\begin{aligned}
&\text{1. } x - 2y + 3z = 4 \\
&\text{2. } 2x - 3y + az = 5 \\
&\text{3. } 3x - 4y + 5z = b
\end{aligned}
\]

We need to determine the values of the constants \(a\) and \(b\) for the system to have:
1. No solution.
2. A unique solution.
3. Infinitely many solutions.

### 1. No Solution

For the system to have **no solution**, the determinant of the coefficient matrix should be non-zero, but the system should be inconsistent.

First, let's write down the coefficient matrix \(A\) and the augmented matrix \( [A | \mathbf{B}] \):

\[
A = \begin{pmatrix}
1 & -2 & 3 \\
2 & -3 & a \\
3 & -4 & 5
\end{pmatrix}, \quad
\mathbf{B} = \begin{pmatrix}
4 \\
5 \\
b
\end{pmatrix}
\]

To find when the system has no solution, we first check the determinant of the coefficient matrix:

\[
\text{Det}(A) = \begin{vmatrix}
1 & -2 & 3 \\
2 & -3 & a \\
3 & -4 & 5
\end{vmatrix}
\]

Expanding this determinant along the first row:

\[
\text{Det}(A) = 1 \cdot \begin{vmatrix} -3 & a \\ -4 & 5 \end{vmatrix} - (-2) \cdot \begin{vmatrix} 2 & a \\ 3 & 5 \end{vmatrix} + 3 \cdot \begin{vmatrix} 2 & -3 \\ 3 & -4 \end{vmatrix}
\]

This simplifies to:

\[
\text{Det}(A) = 1 \cdot (-3(5) - (-4)a) + 2 \cdot (2(5) - 3a) + 3 \cdot (2(-4) - (-3)3)
\]
\[
\text{Det}(A) = 1 \cdot (-15 + 4a) + 2 \cdot (10 - 3a) + 3 \cdot (-8 + 9)
\]
\[
\text{Det}(A) = -15 + 4a + 20 - 6a + 3 \cdot 1
\]
\[
\text{Det}(A) = 8 - 2a
\]

For the system to have no solution, the determinant should not be equal to zero (to avoid having infinite solutions), but the augmented matrix should be inconsistent.

Since \(\text{Det}(A) = 8 - 2a\), for no solution, \(a \neq 4\).

Now, if \(a \neq 4\), \(\text{Det}(A) \neq 0\), and for the system to be inconsistent, we need the augmented matrix to not be consistent. For that, \(b\) should be chosen so that after Gaussian elimination, the system is inconsistent.

**In summary:** \(a \neq 4\) and \(b\) should be chosen so that the system is inconsistent after substitution.

### 2. Unique Solution

For the system to have a **unique solution**, the determinant of the coefficient matrix must be non-zero. Hence, for the system to have a unique solution:

\[
8 - 2a \neq 0 \implies a \neq 4
\]

So, for any value of \(a\) not equal to 4, the system will have a unique solution, and \(b\) can be any real number.

**In summary:** \(a \neq 4\) and \(b\) can be any real number.

### 3. Infinitely Many Solutions

For the system to have **infinitely many solutions**, the determinant of the coefficient matrix must be zero, and the augmented matrix must be consistent.

\[
8 - 2a = 0 \implies a = 4
\]

Now, with \(a = 4\), the determinant of the coefficient matrix is zero. For infinitely many solutions, \(b\) must be such that the system of equations is consistent.

Substitute \(a = 4\) into the system:

\[
\begin{aligned}
&x - 2y + 3z = 4 \\
&2x - 3y + 4z = 5 \\
&3x - 4y + 5z = b
\end{aligned}
\]

This system must have consistent rows. After substituting \(a = 4\) and performing Gaussian elimination, the third equation should not introduce any inconsistency, i.e., \(b\) should satisfy the relation obtained from the consistency condition of the augmented matrix.

**In summary:** \(a = 4\) and \(b\) should satisfy the consistency condition obtained from the system after substitution.

---

Would you like details on any of these points or have any other questions? Here are some related questions to consider:

1. How to perform Gaussian elimination to check for inconsistency?
2. What is the role of the determinant in determining the nature of solutions?
3. How do augmented matrices help in determining the consistency of systems?
4. Can we have cases where the system is underdetermined and still has a unique solution?
5. What happens if all coefficients are zero in a system of linear equations?

**Tip:** Always check the determinant of the coefficient matrix first to quickly assess the nature of the solutions of the system.

Explore how to determine the values of constants a and b in a system of linear equations to achieve no solution, unique solution, or infinitely many solutions. This problem involves linear algebra concepts such as determinants and Gaussian elimination, suitable for advanced high school students.

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