Find the equation to the straight line passing 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 passing through the point (2, -9) and the intersection of the lines given by the equations \(2x + 5y - 8 = 0\) and \(3x - 4y = 35\), we will follow these steps:

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

To find the intersection point of the lines \(2x + 5y - 8 = 0\) and \(3x - 4y = 35\), we will solve these equations simultaneously.

1. Rearranging the first equation:
\[ 2x + 5y = 8 \quad \text{(Equation 1)} \]

2. Rearranging the second equation:
\[ 3x - 4y = 35 \quad \text{(Equation 2)} \]

Now, we can express \(x\) from Equation 1:
\[ x = \frac{8 - 5y}{2} \]

Substituting this expression for \(x\) into Equation 2:
\[ 3\left(\frac{8 - 5y}{2}\right) - 4y = 35 \]

### Step 2: Solve for \(y\)

Multiplying through by 2 to eliminate the fraction:
\[ 3(8 - 5y) - 8y = 70 \]
\[ 24 - 15y - 8y = 70 \]
\[ 24 - 23y = 70 \]
\[ -23y = 70 - 24 \]
\[ -23y = 46 \]
\[ y = -2 \]

### Step 3: Substitute \(y\) back to find \(x\)

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

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

### Step 4: Find the equation of the line through the points (2, -9) and (9, -2)

Now we have two points: \(A(2, -9)\) and \(B(9, -2)\). We can find the slope \(m\) of the line passing through these points:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - (-9)}{9 - 2} = \frac{7}{7} = 1 \]

### Step 5: Use point-slope form to find the equation of the line

Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Using point \(A(2, -9)\):
\[ y - (-9) = 1(x - 2) \]
\[ y + 9 = x - 2 \]
\[ y = x - 2 - 9 \]
\[ y = x - 11 \]

### Final Equation

Rearranging gives us the final equation of the line:
\[ x - y - 11 = 0 \]