Determine the values of ‘a’ and ‘b’ so that the following system of equations has (i) a unique solution, (ii) many solutions, (iii) no solution respectively. 2x + 3y + z = 8 x + 5y + az = 3	 3x + 4y + 7z = b	 Hence find the solutions for cases (i) and (ii)


We are given the following system of linear equations:

1. \( 2x + 3y + z = 8 \)
2. \( x + 5y + az = 3 \)
3. \( 3x + 4y + 7z = b \)

We need to determine the values of \(a\) and \(b\) such that:
- (i) The system has a **unique solution**.
- (ii) The system has **many solutions**.
- (iii) The system has **no solution**.

### Step 1: Write the system as a matrix
We can represent the system of equations in matrix form as:

\[
\begin{bmatrix}
2 & 3 & 1 \\
1 & 5 & a \\
3 & 4 & 7
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
z
\end{bmatrix}
=
\begin{bmatrix}
8 \\
3 \\
b
\end{bmatrix}
\]

This is a system of linear equations, and the matrix form of this system is \(A \mathbf{x} = \mathbf{b}\), where:

\[
A = \begin{bmatrix}
2 & 3 & 1 \\
1 & 5 & a \\
3 & 4 & 7
\end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 8 \\ 3 \\ b \end{bmatrix}
\]

### Step 2: Conditions for a unique solution, many solutions, and no solution
The determinant of the coefficient matrix \(A\) plays a key role in determining whether the system has a unique solution, many solutions, or no solution.

#### (i) **Unique solution**: 
For a unique solution, the system will have a non-zero determinant. Thus, we compute the determinant of matrix \(A\):

\[
\text{det}(A) = \begin{vmatrix}
2 & 3 & 1 \\
1 & 5 & a \\
3 & 4 & 7
\end{vmatrix}
\]

We calculate this determinant using cofactor expansion:

\[
\text{det}(A) = 2 \begin{vmatrix} 5 & a \\ 4 & 7 \end{vmatrix} - 3 \begin{vmatrix} 1 & a \\ 3 & 7 \end{vmatrix} + 1 \begin{vmatrix} 1 & 5 \\ 3 & 4 \end{vmatrix}
\]

Calculating the 2x2 determinants:

\[
\begin{vmatrix} 5 & a \\ 4 & 7 \end{vmatrix} = (5)(7) - (a)(4) = 35 - 4a
\]
\[
\begin{vmatrix} 1 & a \\ 3 & 7 \end{vmatrix} = (1)(7) - (a)(3) = 7 - 3a
\]
\[
\begin{vmatrix} 1 & 5 \\ 3 & 4 \end{vmatrix} = (1)(4) - (5)(3) = 4 - 15 = -11
\]

Now substitute back:

\[
\text{det}(A) = 2(35 - 4a) - 3(7 - 3a) + 1(-11)
\]
\[
\text{det}(A) = 70 - 8a - 21 + 9a - 11
\]
\[
\text{det}(A) = 38 + a
\]

For a unique solution, we require:

\[
\text{det}(A) \neq 0 \quad \Rightarrow \quad 38 + a \neq 0 \quad \Rightarrow \quad a \neq -38
\]

Thus, for a unique solution, \(a \neq -38\).

#### (ii) **Many solutions**: 
For many solutions, the system must be **consistent** and the determinant of the coefficient matrix must be zero. This implies that \(a = -38\) and the system must be consistent.

So, for many solutions:

\[
a = -38 \quad \text{and the system must be consistent.}
\]

#### (iii) **No solution**: 
For no solution, the system must be **inconsistent**. This typically happens when the determinant is zero, but the system is not consistent. 

Thus, for no solution, we would also have \(a = -38\), but the augmented matrix formed by appending \(\mathbf{b}\) to \(A\) must have a different rank than \(A\). This would typically mean the equations represent parallel planes or lines that never intersect. We can investigate this further with the specific values of \(b\).

### Step 3: Finding the solutions for (i) and (ii)

For **unique solutions** (\(a \neq -38\)):
- We solve the system using methods like Gaussian elimination or matrix inversion.

For **many solutions** (\(a = -38\)):
- We can solve the system to find the values of \(b\) that give infinitely many solutions.

Would you like to proceed with solving the system for specific values of \(a\) and \(b\) in cases (i) and (ii)?

Determine the values of ‘a’ and ‘b’ so that the following system of equations has (i) a unique solution, (ii) many solutions, (iii) no solution respectively. 2x + 3y + z = 8, x + 5y + az = 3, 3x + 4y + 7z = b. Hence find the solutions for cases (i) and (ii).

Learn how to determine the values of ‘a’ and ‘b’ in a system of linear equations to achieve a unique solution, many solutions, or no solution. This detailed solution involves matrix determinants, Gaussian elimination, and matrix inversion. Suitable for advanced high school students in Grades 11-12.

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