The point of intersection of the straight lines :
`x-4y=3` and `6x-y=11` is ..........................

Answer

To find the point of intersection of the lines given by the equations \( x - 4y = 3 \) and \( 6x - y = 11 \), we will use the elimination method. Here’s a step-by-step solution:

### Step 1: Write down the equations
We have the two equations:
1. \( x - 4y = 3 \) (Equation 1)
2. \( 6x - y = 11 \) (Equation 2)

### Step 2: Rearrange Equation 1
We can rearrange Equation 1 to express \( x \) in terms of \( y \):
\[ x = 4y + 3 \]

### Step 3: Substitute \( x \) in Equation 2
Now, substitute \( x \) from Equation 1 into Equation 2:
\[ 6(4y + 3) - y = 11 \]

### Step 4: Simplify the equation
Expanding the left side:
\[ 24y + 18 - y = 11 \]
Combine like terms:
\[ 23y + 18 = 11 \]

### Step 5: Solve for \( y \)
Now, isolate \( y \):
\[ 23y = 11 - 18 \]
\[ 23y = -7 \]
\[ y = -\frac{7}{23} \]

### Step 6: Substitute \( y \) back to find \( x \)
Now that we have \( y \), substitute it back into the rearranged Equation 1 to find \( x \):
\[ x = 4\left(-\frac{7}{23}\right) + 3 \]
\[ x = -\frac{28}{23} + 3 \]
Convert 3 to a fraction:
\[ x = -\frac{28}{23} + \frac{69}{23} \]
Combine the fractions:
\[ x = \frac{69 - 28}{23} \]
\[ x = \frac{41}{23} \]

### Step 7: Write the point of intersection
The point of intersection of the two lines is:
\[ \left( \frac{41}{23}, -\frac{7}{23} \right) \]

### Final Answer
The point of intersection is \( \left( \frac{41}{23}, -\frac{7}{23} \right) \).

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