is the following system of linear equations consistent/inconsistent ?Give reason .
`3x +2y = 11`
`9x + 6y = 7`

Answer

To determine whether the given system of linear equations is consistent or inconsistent, we will analyze the equations step by step.

Given equations:
1. \( 3x + 2y = 11 \) (Equation 1)
2. \( 9x + 6y = 7 \) (Equation 2)

### Step 1: Identify coefficients
From the equations, we can identify the coefficients:
- For Equation 1:
- \( a_1 = 3 \)
- \( b_1 = 2 \)
- \( c_1 = -11 \) (Rearranging \( 3x + 2y - 11 = 0 \))

- For Equation 2:
- \( a_2 = 9 \)
- \( b_2 = 6 \)
- \( c_2 = -7 \) (Rearranging \( 9x + 6y - 7 = 0 \))

### Step 2: Calculate ratios
Now we will calculate the ratios of the coefficients:
- \( \frac{a_1}{a_2} = \frac{3}{9} = \frac{1}{3} \)
- \( \frac{b_1}{b_2} = \frac{2}{6} = \frac{1}{3} \)
- \( \frac{c_1}{c_2} = \frac{-11}{-7} = \frac{11}{7} \)

### Step 3: Analyze the conditions
Now we will analyze the conditions for consistency:
- For the system to be consistent, there are two cases:
1. Unique solution: \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \)
2. Infinite solutions: \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \)

- For the system to be inconsistent:
- \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \) but \( \frac{c_1}{c_2} \) is not equal to these ratios.

### Step 4: Conclusion
From our calculations:
- \( \frac{a_1}{a_2} = \frac{1}{3} \)
- \( \frac{b_1}{b_2} = \frac{1}{3} \)
- \( \frac{c_1}{c_2} = \frac{11}{7} \)

Since \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \) but \( \frac{c_1}{c_2} \) is not equal to these ratios, we conclude that the system of equations is inconsistent.

### Final Answer
The system of linear equations is inconsistent.

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