Find the equation of the line passing through the point of intersection of `7x + 6y = 71` and `5x - 8y = -23` , and perpendicular to the line  `4x - 2y = 1`. 

Answer

To find the equation of the line passing through the point of intersection of the lines \(7x + 6y = 71\) and \(5x - 8y = -23\), and perpendicular to the line \(4x - 2y = 1\), we will follow these steps:

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

We have the equations:
1. \(7x + 6y = 71\) (Equation 1)
2. \(5x - 8y = -23\) (Equation 2)

To find the intersection, we can use the method of elimination. We will multiply Equation 1 by 5 and Equation 2 by 7 to eliminate \(x\):

\[ 5(7x + 6y) = 5(71) \implies 35x + 30y = 355 \quad \text{(Equation 3)} \]
\[ 7(5x - 8y) = 7(-23) \implies 35x - 56y = -161 \quad \text{(Equation 4)} \]

Now, we will subtract Equation 4 from Equation 3:

\[ (35x + 30y) - (35x - 56y) = 355 - (-161) \]
\[ 30y + 56y = 355 + 161 \]
\[ 86y = 516 \]
\[ y = \frac{516}{86} = 6 \]

### Step 2: Substitute \(y\) back to find \(x\).

Now that we have \(y = 6\), we can substitute this value back into either of the original equations to find \(x\). We'll use Equation 1:

\[ 7x + 6(6) = 71 \]
\[ 7x + 36 = 71 \]
\[ 7x = 71 - 36 \]
\[ 7x = 35 \]
\[ x = \frac{35}{7} = 5 \]

Thus, the point of intersection is \((5, 6)\).

### Step 3: Find the slope of the line \(4x - 2y = 1\).

We need to find the slope of the line given by \(4x - 2y = 1\). Rearranging this into slope-intercept form \(y = mx + c\):

\[ -2y = -4x + 1 \]
\[ y = 2x - \frac{1}{2} \]

The slope \(m\) of this line is \(2\).

### Step 4: Find the slope of the perpendicular line.

The slope of the line we are looking for, which is perpendicular to this line, can be found using the negative reciprocal of the slope:

\[ m_1 \cdot m_2 = -1 \implies m_2 = -\frac{1}{m_1} = -\frac{1}{2} \]

### Step 5: Write the equation of the line using point-slope form.

Now we have the slope of the required line \(m = -\frac{1}{2}\) and the point \((5, 6)\). We can use the point-slope form of the equation of a line:

\[ y - y_1 = m(x - x_1) \]
Substituting the values:

\[ y - 6 = -\frac{1}{2}(x - 5) \]

### Step 6: Simplify the equation.

Now we simplify this equation:

\[ y - 6 = -\frac{1}{2}x + \frac{5}{2} \]
\[ y = -\frac{1}{2}x + \frac{5}{2} + 6 \]
\[ y = -\frac{1}{2}x + \frac{5}{2} + \frac{12}{2} \]
\[ y = -\frac{1}{2}x + \frac{17}{2} \]

To convert this into standard form, we can multiply through by 2 to eliminate the fraction:

\[ 2y = -x + 17 \]
\[ x + 2y = 17 \]

Thus, the equation of the required line is:

\[ \boxed{x + 2y = 17} \]