Equations of line which passes through the point of intersection of the `4x-3y-1=0` and `2x-5y+3=0` and are equally inclined to the axes are:

Answer

To find the equations of the line that passes through the point of intersection of the lines \(4x - 3y - 1 = 0\) and \(2x - 5y + 3 = 0\) and is equally inclined to the axes, we will follow these steps:

### Step 1: Find the point of intersection of the two lines.

We have the equations:
1. \(4x - 3y - 1 = 0\) (Equation 1)
2. \(2x - 5y + 3 = 0\) (Equation 2)

To find the intersection, we can solve these equations simultaneously.

From Equation 1, we can express \(y\) in terms of \(x\):
\[ 3y = 4x - 1 \implies y = \frac{4x - 1}{3} \tag{1} \]

Substituting Equation (1) into Equation 2:
\[ 2x - 5\left(\frac{4x - 1}{3}\right) + 3 = 0 \]
Multiply through by 3 to eliminate the fraction:
\[ 6x - 5(4x - 1) + 9 = 0 \]
Expanding this:
\[ 6x - 20x + 5 + 9 = 0 \implies -14x + 14 = 0 \]
Solving for \(x\):
\[ -14x = -14 \implies x = 1 \]

Now substituting \(x = 1\) back into Equation (1) to find \(y\):
\[ y = \frac{4(1) - 1}{3} = \frac{3}{3} = 1 \]

Thus, the point of intersection is \((1, 1)\).

### Step 2: Determine the slope of the lines that are equally inclined to the axes.

Lines that are equally inclined to the axes have slopes of \(m = 1\) or \(m = -1\).

### Step 3: Write the equations of the lines using the point-slope form.

Using the point-slope form of the equation of a line, \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the point of intersection \((1, 1)\):

1. For \(m = 1\):
\[ y - 1 = 1(x - 1) \implies y - 1 = x - 1 \implies y = x \]

2. For \(m = -1\):
\[ y - 1 = -1(x - 1) \implies y - 1 = -x + 1 \implies y = -x + 2 \]

### Final Answer:
The equations of the lines that pass through the point of intersection and are equally inclined to the axes are:
1. \(y = x\)
2. \(y = -x + 2\)

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