what is an equation of the line that

The required equation of straight line is $4y = x + 33$

What is slope-intercept form ?

One form of the equation of a straight line is called the slope-intercept form because it contains information about these two properties. In the equation y = mx + c the value of m is called the slope, (or gradient), of the line. It can be positive, negative or zero. The value of c is called the y-intercept which means the point of intersection of the graph of straight line with y axis.

What is the concept of parallel and perpendicular line ?

If we have to find a line parallel to a given straight line, then we take the same positive slope value of the given line for finding the required equation.

And if we If we have to find a line perpendicular to a given straight line, then we take the negative reciprocal slope value of the given line for finding the required equation.

Finding the required equation of straight line in slope-intercept form -

Given equation is   $y - 3 = -4(x + 2)$

$\Rightarrow y - 3 = -4x -8$

$\Rightarrow y = -4x - 5$

The slope of the given line is -4 when comparing with $y = mx + c.$

Thus the required slope of straight line is say $m_1 = -(-1/4) = 1/4$

as the required line is perpendicular with the line.

The equation of straight line is y = m1x + c

$\Rightarrow y = (x/4) + c$

Putting the values of the points (-5,7) in x and y coordinates.

$\Rightarrow 7 = (-5/4) + c$

$\Rightarrow c = 7 + 5/4 = 33/4$

Putting value of y-intercept in original equation,

$\Rightarrow y = (x/4) + 33/4$

$\therefore$  4y = x + 33 which is the required equation.