Given a system of equation in` x,y,z:x+y+z=6,x+2y+3z=10` and `x+2y+az b` . If this system has infinite number of solution , then `a^2 + b = ………`

Explanation

To solve the given system of equations and find the value of \( a^2 + b \) such that the system has an infinite number of solutions, we can follow these steps:

### Step 1: Write down the equations
The given equations are:
1. \( x + y + z = 6 \) (Equation 1)
2. \( x + 2y + 3z = 10 \) (Equation 2)
3. \( x + 2y + az = b \) (Equation 3)

### Step 2: Eliminate one variable
We can express \( y \) and \( z \) in terms of \( x \) using the first two equations.

From Equation 1:
\[ y = 6 - x - z \]

Substituting \( y \) in Equation 2:
\[ x + 2(6 - x - z) + 3z = 10 \]
Expanding this gives:
\[ x + 12 - 2x - 2z + 3z = 10 \]
Simplifying:
\[ -x + z + 12 = 10 \]
Thus,
\[ z - x = -2 \quad \Rightarrow \quad z = x - 2 \quad \text{(Equation 4)} \]

### Step 3: Substitute back into the first equation
Now substitute \( z \) back into Equation 1:
\[ x + y + (x - 2) = 6 \]
This simplifies to:
\[ 2x + y - 2 = 6 \quad \Rightarrow \quad y = 8 - 2x \quad \text{(Equation 5)} \]

### Step 4: Substitute \( y \) and \( z \) into Equation 3
Now substitute \( y \) and \( z \) from Equations 4 and 5 into Equation 3:
\[ x + 2(8 - 2x) + a(x - 2) = b \]
Expanding this gives:
\[ x + 16 - 4x + ax - 2a = b \]
Combining like terms:
\[ (1 - 4 + a)x + (16 - 2a) = b \]
This simplifies to:
\[ (a - 3)x + (16 - 2a) = b \]

### Step 5: Set conditions for infinite solutions
For the system to have an infinite number of solutions, the coefficients of \( x \) must be zero and the constant terms must be equal:
1. \( a - 3 = 0 \) (Coefficient of \( x \))
2. \( 16 - 2a = b \) (Constant term)

From the first condition:
\[ a = 3 \]

Substituting \( a = 3 \) into the second condition:
\[ 16 - 2(3) = b \quad \Rightarrow \quad 16 - 6 = b \quad \Rightarrow \quad b = 10 \]

### Step 6: Calculate \( a^2 + b \)
Now we can find \( a^2 + b \):
\[ a^2 + b = 3^2 + 10 = 9 + 10 = 19 \]

### Final Answer
Thus, the value of \( a^2 + b \) is:
\[ \boxed{19} \]