Find the equation of the straight line which passes through
the point (2, -9) and the intersection of the lines `2x+5y-8=0 and 3x-4y=35`

Answer

To find the equation of the straight line that passes through the point (2, -9) and the intersection of the lines \(2x + 5y - 8 = 0\) and \(3x - 4y - 35 = 0\), we will follow these steps:

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

We have the equations of the two lines:
1. \(2x + 5y - 8 = 0\) (Equation 1)
2. \(3x - 4y - 35 = 0\) (Equation 2)

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

From Equation 1, we can express \(y\) in terms of \(x\):
\[ 5y = 8 - 2x \implies y = \frac{8 - 2x}{5} \]

Now, substitute this expression for \(y\) into Equation 2:
\[ 3x - 4\left(\frac{8 - 2x}{5}\right) - 35 = 0 \]
Multiply through by 5 to eliminate the fraction:
\[ 15x - 4(8 - 2x) - 175 = 0 \]
Distributing the \(-4\):
\[ 15x - 32 + 8x - 175 = 0 \]
Combine like terms:
\[ 23x - 207 = 0 \]
Solving for \(x\):
\[ 23x = 207 \implies x = \frac{207}{23} = 9 \]

Now substitute \(x = 9\) back into the expression for \(y\):
\[ y = \frac{8 - 2(9)}{5} = \frac{8 - 18}{5} = \frac{-10}{5} = -2 \]

Thus, the intersection point is \((9, -2)\).

### Step 2: Find the equation of the line through the points (2, -9) and (9, -2)

We can use the two-point form of the equation of a line. The slope \(m\) between the points \((x_1, y_1) = (2, -9)\) and \((x_2, y_2) = (9, -2)\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - (-9)}{9 - 2} = \frac{7}{7} = 1 \]

Now, using the point-slope form of the line equation:
\[ y - y_1 = m(x - x_1) \]
Substituting \(m = 1\), \(x_1 = 2\), and \(y_1 = -9\):
\[ y - (-9) = 1(x - 2) \]
Simplifying this:
\[ y + 9 = x - 2 \]
\[ y = x - 11 \]

### Final Equation

The equation of the line is:
\[ x - y - 11 = 0 \]