Check whether the given system of equations has Unique solution, No solution or Infinite solutions: `{:(x - 3y = 3),(3x - 9y = 2):}`

Answer

To determine whether the given system of equations has a unique solution, no solution, or infinite solutions, we will analyze the equations step by step.

**Given Equations:**
1. \( x - 3y = 3 \) (Equation 1)
2. \( 3x - 9y = 2 \) (Equation 2)

### Step 1: Rewrite the equations in standard form
We can rewrite both equations in the standard form \( ax + by + c = 0 \).

- For Equation 1:
\[ x - 3y - 3 = 0 \implies 1x - 3y + (-3) = 0 \]
Here, \( a_1 = 1, b_1 = -3, c_1 = -3 \).

- For Equation 2:
\[ 3x - 9y - 2 = 0 \implies 3x - 9y + (-2) = 0 \]
Here, \( a_2 = 3, b_2 = -9, c_2 = -2 \).

### Step 2: Calculate the ratios
Next, we will calculate the ratios \( \frac{a_1}{a_2}, \frac{b_1}{b_2}, \) and \( \frac{c_1}{c_2} \).

- Calculate \( \frac{a_1}{a_2} \):
\[ \frac{a_1}{a_2} = \frac{1}{3} \]

- Calculate \( \frac{b_1}{b_2} \):
\[ \frac{b_1}{b_2} = \frac{-3}{-9} = \frac{1}{3} \]

- Calculate \( \frac{c_1}{c_2} \):
\[ \frac{c_1}{c_2} = \frac{-3}{-2} = \frac{3}{2} \]

### Step 3: Analyze the ratios
Now we will analyze the results of the ratios:

- We found:
\[ \frac{a_1}{a_2} = \frac{1}{3}, \quad \frac{b_1}{b_2} = \frac{1}{3}, \quad \frac{c_1}{c_2} = \frac{3}{2} \]

### Step 4: Determine the type of solution
According to the conditions for the types of solutions:

1. **Unique solution**: If \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \)
2. **No solution**: If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \) but \( \frac{c_1}{c_2} \neq \frac{a_1}{a_2} \)
3. **Infinite solutions**: If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \)

From our calculations:
- \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \) (both equal to \( \frac{1}{3} \))
- \( \frac{c_1}{c_2} \) is not equal to \( \frac{a_1}{a_2} \) (since \( \frac{3}{2} \neq \frac{1}{3} \))

Thus, the system of equations satisfies the condition for **no solution**.

### Conclusion
The given system of equations has **no solution**.

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