Find the equation of a line passing through the point of intersection of the lines `x+y=4` and `2x-3y-1=0` and parallel to a line whose intercepts on the axes are `4` and `6` units.

Answer

To find the equation of the line passing through the point of intersection of the lines \(x + y = 4\) and \(2x - 3y - 1 = 0\) and parallel to a line whose intercepts on the axes are \(4\) and \(6\) units, we can follow these steps:

### Step 1: Find the Point of Intersection of the Given Lines
We have two equations:
1. \(x + y = 4\) (Equation 1)
2. \(2x - 3y - 1 = 0\) (Equation 2)

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

From Equation 1, we can express \(y\) in terms of \(x\):
\[ y = 4 - x \]

Now substitute \(y\) in Equation 2:
\[ 2x - 3(4 - x) - 1 = 0 \]
\[ 2x - 12 + 3x - 1 = 0 \]
\[ 5x - 13 = 0 \]
\[ x = \frac{13}{5} \]

Now substitute \(x\) back into Equation 1 to find \(y\):
\[ y = 4 - \frac{13}{5} = \frac{20}{5} - \frac{13}{5} = \frac{7}{5} \]

Thus, the point of intersection is:
\[ \left(\frac{13}{5}, \frac{7}{5}\right) \]

### Step 2: Find the Equation of the Line with Given Intercepts
The line whose intercepts on the axes are \(4\) and \(6\) can be expressed using the intercept form of a line:
\[ \frac{x}{4} + \frac{y}{6} = 1 \]

To convert this into standard form:
\[ 6x + 4y = 24 \]
or simplified:
\[ 3x + 2y = 12 \]

### Step 3: Find the Slope of the Given Line
The slope \(m\) of the line \(3x + 2y = 12\) can be found by rearranging it into slope-intercept form \(y = mx + b\):
\[ 2y = -3x + 12 \implies y = -\frac{3}{2}x + 6 \]
Thus, the slope \(m = -\frac{3}{2}\).

### Step 4: Write the Equation of the Required Line
Since the required line is parallel to the line we just found, it will have the same slope \(m = -\frac{3}{2}\). We can use the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Substituting \(m = -\frac{3}{2}\) and the point of intersection \(\left(\frac{13}{5}, \frac{7}{5}\right)\):
\[ y - \frac{7}{5} = -\frac{3}{2}\left(x - \frac{13}{5}\right) \]

### Step 5: Simplify the Equation
Multiply through by \(10\) to eliminate the fractions:
\[ 10\left(y - \frac{7}{5}\right) = -15\left(x - \frac{13}{5}\right) \]
\[ 10y - 14 = -15x + 39 \]
Rearranging gives:
\[ 15x + 10y = 53 \]

### Final Answer
The equation of the required line is:
\[ 15x + 10y = 53 \]