Find the equation of a line passing through the point of intersection of the lines `3x+5y=-2` and `5x-2y=7` and perpendicular to the line `4x-5y+1=0`.

Answer

To find the equation of a line passing through the point of intersection of the lines \(3x + 5y = -2\) and \(5x - 2y = 7\), and perpendicular to the line \(4x - 5y + 1 = 0\), we can follow these steps:

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

We have the equations:
1. \(3x + 5y = -2\) (Equation 1)
2. \(5x - 2y = 7\) (Equation 2)

To find the intersection, we can solve these equations simultaneously. We will eliminate one variable by manipulating the equations.

Multiply Equation 1 by 2:
\[ 2(3x + 5y) = 2(-2) \implies 6x + 10y = -4 \quad \text{(Equation 3)} \]

Multiply Equation 2 by 5:
\[ 5(5x - 2y) = 5(7) \implies 25x - 10y = 35 \quad \text{(Equation 4)} \]

Now, we can add Equation 3 and Equation 4:
\[ (6x + 10y) + (25x - 10y) = -4 + 35 \]
\[ 31x = 31 \implies x = 1 \]

Now substitute \(x = 1\) back into Equation 1 to find \(y\):
\[ 3(1) + 5y = -2 \implies 3 + 5y = -2 \implies 5y = -5 \implies y = -1 \]

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

### Step 2: Find the slope of the given line.

The equation of the line we want to be perpendicular to is:
\[ 4x - 5y + 1 = 0 \]
We can rewrite this in slope-intercept form \(y = mx + c\):
\[ -5y = -4x - 1 \implies y = \frac{4}{5}x + \frac{1}{5} \]
The slope \(m_1\) of this line is \(\frac{4}{5}\).

### Step 3: Find the slope of the perpendicular line.

The slope \(m_2\) of the line perpendicular to it is given by the negative reciprocal:
\[ m_2 = -\frac{1}{m_1} = -\frac{1}{\frac{4}{5}} = -\frac{5}{4} \]

### Step 4: Use the point-slope form to find the equation of the required line.

Using the point-slope form of the line equation:
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1) = (1, -1)\) and \(m = -\frac{5}{4}\):
\[ y - (-1) = -\frac{5}{4}(x - 1) \]
\[ y + 1 = -\frac{5}{4}x + \frac{5}{4} \]
\[ y = -\frac{5}{4}x + \frac{5}{4} - 1 \]
\[ y = -\frac{5}{4}x + \frac{5}{4} - \frac{4}{4} \]
\[ y = -\frac{5}{4}x + \frac{1}{4} \]

### Step 5: Convert to standard form.

To convert this equation to standard form \(Ax + By + C = 0\):
\[ \frac{5}{4}x + y - \frac{1}{4} = 0 \]
Multiply through by 4 to eliminate the fraction:
\[ 5x + 4y - 1 = 0 \]

Thus, the equation of the required line is:
\[ 5x + 4y - 1 = 0 \]