Find the equation of the line passing through the intersection of the lines `3x-4y+1=0` and `5x+y-1=0` which cuts off equal intercepts on the axes.

Answer

To find the equation of the line passing through the intersection of the lines \(3x - 4y + 1 = 0\) and \(5x + y - 1 = 0\) which cuts off equal intercepts on the axes, we can follow these steps:

### Step 1: Find the intersection point of the two lines.
To find the intersection of the lines, we can solve the two equations simultaneously.

1. The first equation is:
\[ 3x - 4y + 1 = 0 \quad \text{(1)} \]

2. The second equation is:
\[ 5x + y - 1 = 0 \quad \text{(2)} \]

From equation (2), we can express \(y\) in terms of \(x\):
\[ y = 1 - 5x \quad \text{(3)} \]

### Step 2: Substitute equation (3) into equation (1).
Substituting \(y\) from equation (3) into equation (1):
\[ 3x - 4(1 - 5x) + 1 = 0 \]
Expanding this:
\[ 3x - 4 + 20x + 1 = 0 \]
Combining like terms:
\[ 23x - 3 = 0 \]
Solving for \(x\):
\[ x = \frac{3}{23} \]

### Step 3: Find the corresponding \(y\) value.
Substituting \(x = \frac{3}{23}\) back into equation (3):
\[ y = 1 - 5\left(\frac{3}{23}\right) = 1 - \frac{15}{23} = \frac{8}{23} \]

### Step 4: Write the coordinates of the intersection point.
The intersection point of the two lines is:
\[ \left(\frac{3}{23}, \frac{8}{23}\right) \]

### Step 5: Formulate the equation of the line with equal intercepts.
The general form of the equation of a line that cuts equal intercepts on the axes is given by:
\[ \frac{x}{a} + \frac{y}{a} = 1 \]
This simplifies to:
\[ x + y = a \]

### Step 6: Determine the value of \(a\).
Since the line passes through the point \(\left(\frac{3}{23}, \frac{8}{23}\right)\), we substitute these values into the equation:
\[ \frac{3}{23} + \frac{8}{23} = a \]
Calculating \(a\):
\[ a = \frac{11}{23} \]

### Step 7: Write the final equation of the line.
Substituting \(a\) back into the equation:
\[ x + y = \frac{11}{23} \]
To eliminate the fraction, we can multiply through by 23:
\[ 23x + 23y = 11 \]

### Final Answer:
The equation of the line is:
\[ 23x + 23y = 11 \]