Find the equation to the straight line
passing through the point `(- 6, 10)` and perpendicular to the straight line `7x+ 8y= 5`.

Answer

To find the equation of the straight line passing through the point \((-6, 10)\) and perpendicular to the line given by the equation \(7x + 8y = 5\), we can follow these steps:

### Step 1: Find the slope of the given line
The first step is to convert the equation of the line \(7x + 8y = 5\) into slope-intercept form \(y = mx + c\), where \(m\) is the slope.

1. Rearranging the equation:
\[ 8y = -7x + 5 \]
\[ y = -\frac{7}{8}x + \frac{5}{8} \]

2. From this, we can see that the slope \(m_1\) of the given line is:
\[ m_1 = -\frac{7}{8} \]

### Step 2: Find the slope of the perpendicular line
The slope of the line perpendicular to another line is the negative reciprocal of the slope of the original line. Therefore, if \(m_1 = -\frac{7}{8}\), the slope \(m_2\) of the perpendicular line is:
\[ m_2 = -\frac{1}{m_1} = -\frac{1}{-\frac{7}{8}} = \frac{8}{7} \]

### Step 3: Use the point-slope form to find the equation of the line
Now that we have the slope of the perpendicular line, we can use the point-slope form of the equation of a line, which is given by:
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1)\) is the point through which the line passes, and \(m\) is the slope.

Substituting \((x_1, y_1) = (-6, 10)\) and \(m = \frac{8}{7}\):
\[ y - 10 = \frac{8}{7}(x + 6) \]

### Step 4: Simplify the equation
Now, we will simplify the equation:
1. Distributing the slope:
\[ y - 10 = \frac{8}{7}x + \frac{48}{7} \]

2. Adding 10 (which is \(\frac{70}{7}\)) to both sides:
\[ y = \frac{8}{7}x + \frac{48}{7} + \frac{70}{7} \]
\[ y = \frac{8}{7}x + \frac{118}{7} \]

### Final Equation
Thus, the equation of the line passing through the point \((-6, 10)\) and perpendicular to the line \(7x + 8y = 5\) is:
\[ y = \frac{8}{7}x + \frac{118}{7} \]