If the ponts A(3,5) and B(1,4) lies on the graph of line ax +by =7 find the value of a

Answer

To solve the problem, we need to find the value of \( a \) in the equation \( ax + by = 7 \) given that the points \( A(3, 5) \) and \( B(1, 4) \) lie on the line.

### Step-by-Step Solution:

1. **Substituting Point A into the Equation:**
We start by substituting the coordinates of point A(3, 5) into the equation \( ax + by = 7 \).
\[ a(3) + b(5) = 7 \]
This simplifies to:
\[ 3a + 5b = 7 \quad \text{(Equation 1)} \]

2. **Substituting Point B into the Equation:**
Next, we substitute the coordinates of point B(1, 4) into the same equation.
\[ a(1) + b(4) = 7 \]
This simplifies to:
\[ a + 4b = 7 \quad \text{(Equation 2)} \]

3. **Solving the System of Equations:**
Now we have a system of two equations:
- Equation 1: \( 3a + 5b = 7 \)
- Equation 2: \( a + 4b = 7 \)

We can solve these equations simultaneously.

4. **Multiplying Equation 2:**
To eliminate \( a \), we can multiply Equation 2 by 3:
\[ 3(a + 4b) = 3(7) \]
This gives us:
\[ 3a + 12b = 21 \quad \text{(Equation 3)} \]

5. **Subtracting Equations:**
Now we subtract Equation 1 from Equation 3:
\[ (3a + 12b) - (3a + 5b) = 21 - 7 \]
This simplifies to:
\[ 7b = 14 \]
Dividing both sides by 7 gives:
\[ b = 2 \]

6. **Finding the Value of \( a \):**
Now that we have \( b \), we can substitute \( b = 2 \) back into Equation 2 to find \( a \):
\[ a + 4(2) = 7 \]
This simplifies to:
\[ a + 8 = 7 \]
Subtracting 8 from both sides gives:
\[ a = 7 - 8 = -1 \]

### Final Answer:
Thus, the value of \( a \) is \( -1 \).