If the points A(4,6) and B(1,3) lie on the graph of ax +by =8 then find the value of a and b

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) such that the points \( A(4,6) \) and \( B(1,3) \) lie on the line represented by the equation \( ax + by = 8 \).

### Step 1: Substitute Point A into the Equation
We start by substituting the coordinates of point \( A(4, 6) \) into the equation \( ax + by = 8 \).

\[ a(4) + b(6) = 8 \]

This simplifies to:

\[ 4a + 6b = 8 \quad \text{(Equation 1)} \]

### Step 2: Substitute Point B into the Equation
Next, we substitute the coordinates of point \( B(1, 3) \) into the same equation.

\[ a(1) + b(3) = 8 \]

This simplifies to:

\[ a + 3b = 8 \quad \text{(Equation 2)} \]

### Step 3: Solve the System of Equations
Now, we have a system of equations:

1. \( 4a + 6b = 8 \) (Equation 1)
2. \( a + 3b = 8 \) (Equation 2)

We can solve these equations simultaneously. Let's first multiply Equation 2 by 4 to align the coefficients of \( a \):

\[ 4(a + 3b) = 4(8) \]

This gives us:

\[ 4a + 12b = 32 \quad \text{(Equation 3)} \]

### Step 4: Subtract Equation 1 from Equation 3
Now, we subtract Equation 1 from Equation 3:

\[ (4a + 12b) - (4a + 6b) = 32 - 8 \]

This simplifies to:

\[ 6b = 24 \]

### Step 5: Solve for \( b \)
Now, we can solve for \( b \):

\[ b = \frac{24}{6} = 4 \]

### Step 6: Substitute \( b \) Back to Find \( a \)
Now that we have \( b = 4 \), we can substitute this value back into Equation 2 to find \( a \):

\[ a + 3(4) = 8 \]

This simplifies to:

\[ a + 12 = 8 \]

So,

\[ a = 8 - 12 = -4 \]

### Final Values
Thus, the values of \( a \) and \( b \) are:

\[ a = -4, \quad b = 4 \]

### Summary
The final answer is:
- \( a = -4 \)
- \( b = 4 \)