Find the equation of the straight line which passes through the point of intersection of the straight lines 3x-4y +1=0 and 5x+y - 1 =0 and cuts off equal intercepts from the aixs .

Options

A. 32 x + 32y +11 =0
B. 23x + 23y =11
C. 9x+18y +5=0
D. none of these

Answer

To solve the problem step by step, we need to find the equation of the straight line that meets the specified conditions.

### Step 1: Find the point of intersection of the two lines
We have the equations of the two lines:
1. \(3x - 4y + 1 = 0\) (Equation 1)
2. \(5x + y - 1 = 0\) (Equation 2)

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

From Equation 2, we can express \(y\) in terms of \(x\):
\[ y = 1 - 5x \]

Now, substitute this expression for \(y\) into Equation 1:
\[ 3x - 4(1 - 5x) + 1 = 0 \]
\[ 3x - 4 + 20x + 1 = 0 \]
\[ 23x - 3 = 0 \]
\[ 23x = 3 \implies x = \frac{3}{23} \]

Now, substitute \(x = \frac{3}{23}\) back into the expression for \(y\):
\[ y = 1 - 5\left(\frac{3}{23}\right) = 1 - \frac{15}{23} = \frac{8}{23} \]

So, the point of intersection is \(\left(\frac{3}{23}, \frac{8}{23}\right)\).

### Step 2: Equation of the line with equal intercepts
A line that cuts off equal intercepts from the axes can be represented as:
\[ x + y = a \]
where \(a\) is the length of the intercepts on both axes.

### Step 3: Substitute the point of intersection into the line equation
Since the line passes through the point of intersection \(\left(\frac{3}{23}, \frac{8}{23}\right)\), we substitute these coordinates into the equation:
\[ \frac{3}{23} + \frac{8}{23} = a \]
\[ \frac{11}{23} = a \]

### Step 4: Write the equation of the line
Now we can write the equation of the line using the value of \(a\):
\[ x + y = \frac{11}{23} \]

To express this in a standard form, we can multiply through by 23 to eliminate the fraction:
\[ 23x + 23y = 11 \]

### Final Answer
Thus, the equation of the straight line is:
\[ 23x + 23y - 11 = 0 \]