An equation of the line contains a point (-3, 7) and this line is perpendicular to another line \(y = - 4x - 5\). What will be the equation of another line?

Use the standard theory of two lines:

A line contains a point  \(\left( {{x_1},{y_1}} \right)\) and perpendicular to another line 

\(y = mx + c\) , 

then,

\(m \times {m_1} =  - 1\)

Here, \({m_1}\)is the slop of another line.

Then, the equation of line that contains a point is,

            \(y - {y_1} = {m_1}\left( {x - {x_1}} \right)\)

Consider the equation:

           \(y =  - 4x - 5\)

Now, compare with the general equation of a line\(y = mx + c\).

Here,\(m =  - 4\)

Then, the formula of two lines is perpendicular,

           \(m \times {m_1} =  - 1\)

Therefore,

           \(\begin{array}{c} - 4 \times {m_1} =  - 1\\{m_1} = \frac{1}{4}\end{array}\)

Then, the equation of line that contains a point \(\left( { - 3,7} \right)\) is,

           \(\begin{array}{c}y - {y_1} = {m_1}\left( {x - {x_1}} \right)\\y - 7 = \frac{1}{4}\left( {x - \left( { - 3} \right)} \right)\\4\left( {y - 7} \right) = x + 3\\4y - 28 = x + 3\end{array}\)

Therefore,

\(\begin{array}{c}x - 4y + 31 = 0\\y = \frac{1}{4}x + \frac{{31}}{4}\end{array}\)