If the system of equations ax +y=3,x+2y=3 , 3x+4y=7 is consistent then value of a is given by

Options

A. 2
B. 1
C. `-1`
D. 0

Answer

To find the value of \( a \) for which the system of equations is consistent, we will follow these steps:

### Step 1: Write down the equations
We have the following equations:
1. \( ax + y = 3 \) (Equation 1)
2. \( x + 2y = 3 \) (Equation 2)
3. \( 3x + 4y = 7 \) (Equation 3)

### Step 2: Solve Equations 2 and 3
We will solve Equations 2 and 3 using the elimination method.

From Equation 2:
\[ x + 2y = 3 \quad \text{(Multiply this equation by 3)} \]
This gives:
\[ 3x + 6y = 9 \quad \text{(Equation 4)} \]

Now we have:
- Equation 4: \( 3x + 6y = 9 \)
- Equation 3: \( 3x + 4y = 7 \)

### Step 3: Subtract Equation 3 from Equation 4
Now we will subtract Equation 3 from Equation 4:
\[ (3x + 6y) - (3x + 4y) = 9 - 7 \]
This simplifies to:
\[ 2y = 2 \]

### Step 4: Solve for \( y \)
Dividing both sides by 2 gives:
\[ y = 1 \]

### Step 5: Substitute \( y \) back to find \( x \)
Now we will substitute \( y = 1 \) back into Equation 2 to find \( x \):
\[ x + 2(1) = 3 \]
This simplifies to:
\[ x + 2 = 3 \]
Subtracting 2 from both sides gives:
\[ x = 1 \]

### Step 6: Find the point of intersection
The point of intersection of the second and third equations is \( (1, 1) \).

### Step 7: Substitute \( x \) and \( y \) into Equation 1
Now we will substitute \( x = 1 \) and \( y = 1 \) into Equation 1 to find \( a \):
\[ a(1) + 1 = 3 \]
This simplifies to:
\[ a + 1 = 3 \]
Subtracting 1 from both sides gives:
\[ a = 2 \]

### Final Answer
Thus, the value of \( a \) for which the system of equations is consistent is:
\[ \boxed{2} \]