Are the following pair of linear equations consistent? Justify your answer.
`x+ 3y =11, 2(2x+ 6y)=22`

Answer

To determine whether the given pair of linear equations is consistent or inconsistent, we will follow these steps:

### Step 1: Write down the equations
The given equations are:
1. \( x + 3y = 11 \)
2. \( 2(2x + 6y) = 22 \)

### Step 2: Simplify the second equation
We start by simplifying the second equation:
\[ 2(2x + 6y) = 22 \]
Dividing both sides by 2 gives:
\[ 2x + 6y = 11 \]

### Step 3: Rewrite both equations in standard form
Now we have the two equations:
1. \( x + 3y - 11 = 0 \) (Equation 1)
2. \( 2x + 6y - 11 = 0 \) (Equation 2)

### Step 4: Identify coefficients
From the equations, we can identify the coefficients:
- For Equation 1: \( a_1 = 1, b_1 = 3, c_1 = -11 \)
- For Equation 2: \( a_2 = 2, b_2 = 6, c_2 = -11 \)

### Step 5: Check the ratios of coefficients
Now we will check the ratios of the coefficients:
\[ \frac{a_1}{a_2} = \frac{1}{2}, \quad \frac{b_1}{b_2} = \frac{3}{6} = \frac{1}{2}, \quad \frac{c_1}{c_2} = \frac{-11}{-11} = 1 \]

### Step 6: Analyze the ratios
We find that:
- \( \frac{a_1}{a_2} = \frac{1}{2} \)
- \( \frac{b_1}{b_2} = \frac{1}{2} \)
- \( \frac{c_1}{c_2} = 1 \)

Since \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \) but \( \frac{c_1}{c_2} \) is not equal to these, we conclude that the lines represented by the equations are parallel.

### Step 7: Conclusion
Since the lines are parallel, they do not intersect, which means there is no solution to the system of equations. Therefore, the pair of linear equations is inconsistent.

### Final Answer
The given pair of linear equations is inconsistent.

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