Find an equation for the line perpendicular to 3 x + 21 y = 105 and goes through the point ( 8 , − 7 ) . Write your answer in the form y = m x + b .

The line is;

y = 7x - 63

Step-by-step explanation:

In this question, we are asked to find the equation of a line that is perpendicular to a line and passes through a specific point.

Firstly, we write the general equation of a line and define the parameters;

Generally for a line;

y = mx + c where m is the slope and c is the y-intercept

Now let’s consider the first line;

3x + 21y = 105

Let us write this in the form of the General line;

21y = -3x + 105

y = -x/7 + 5 ( I divided through by 21)

What can we identify from here? The slope or gradient of the line is -1/7

Now for a line l2 whose gradient is perpendicular to this line;

m1m2 = -1

What this means is that the product of their slopes is equal to -1

We have identified m1

So m2 will be -1/m1

This is same as m2 = -1/(-1/7) = 7

The gradient of the second line is 7

Now to write the equation of the second line, we use the following relation;

y-y1/x-x1 = m

We use the coordinates of the point through which the line passes as given in the question;

y+7/(x-8) = 7

y + 7 = 7(x-8)

y + 7 = 7x - 56

y = 7x-56-7

y = 7x - 63

the answer for the question that you are asking is 4