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

Explanation

To solve the problem, we need to analyze the given system of equations and determine the conditions under which it has an infinite number of solutions.

### Step-by-Step Solution:

1. **Identify the Given Equations:**
We have the following equations:
\[ \text{(1)} \quad x + y + z = 6 \]
\[ \text{(2)} \quad x + 2y + 3z = 10 \]
\[ \text{(3)} \quad x + 2y + az = b \]

2. **Understand the Condition for Infinite Solutions:**
For a system of equations to have an infinite number of solutions, the third equation must be a linear combination of the first two equations. This means that the coefficients of \(x\), \(y\), and \(z\) in the third equation must be expressible in terms of the coefficients in the first two equations.

3. **Set Up the Coefficient Relationships:**
From equations (1) and (2), we can express the coefficients:
- Coefficient of \(x\) in (1) is 1, and in (2) is 1.
- Coefficient of \(y\) in (1) is 1, and in (2) is 2.
- Coefficient of \(z\) in (1) is 1, and in (2) is 3.

For equation (3) to be consistent with equations (1) and (2), we need:
\[ \frac{1}{1} = \frac{2}{2} = \frac{a}{3} \]

4. **Solve for \(a\):**
From the ratio:
\[ \frac{a}{3} = 1 \implies a = 3 \]

5. **Determine \(b\):**
Now, substituting \(a = 3\) into equation (3):
\[ x + 2y + 3z = b \]
We need this equation to be consistent with the previous equations. We can use equation (2) to find \(b\):
\[ b = 10 \]

6. **Calculate \(a^2 + b\):**
Now that we have \(a = 3\) and \(b = 10\):
\[ a^2 + b = 3^2 + 10 = 9 + 10 = 19 \]

### Final Answer:
\[ a^2 + b = 19 \]