# 52. Which of the following is the equation of the line passing through (-1, 5) and perpendicular to the graph of y = -3x + 7 ? 1. y= 1/3x + 16/3 2. y= 1/3 x +14/3

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y= (1/3)x +16/3.......(1)

y= (1/3)x + 14/3.......(2)

Both lines (1) and (2) are perpendicular to the line y= -3x+7 because the product of the slopes = -1

To verify which line passes through the point (-1,5) , we need to substitute in both lines and see which one verifies the point.

For line (1):

y= (1/3)x +16/3

5= (1/3)(-1) + 16/3

5= -1/3 + 16/3

5= 15/3

5= 5

Then line (1) passes through the point (-1.5)

Now to verify line (2):

y= (1/3)x + 14/3

5= (1/3)(-1) + 14/3

5= -1/3 + 14/3

5= 13/3

Then line (2) does not verify the point (-1.5) then it does not pass through it.

**Then the answer is number (1)**

We shall see which of the equation is satisfied by the point (-1,5) and is perpendicular ty =-3x+7.

Solution:

A line perpendicular to the line y = -3x+7 .....(1) should be of the form y = (1/3)x+k, where the coefficient of x is -1 times the coefficient x in the given line y = -3x+7 at (1)

Now since the line y = (1/3)x+k......(2) passes through (-1,5),

5=(1/3)*1+k . Or k = 5-1/3 = 14/3. Substitute this value of k in (2) and we get the required line y= (1/3)x + 14/5 as at choice number 2.

Therefore y = 1/3x+14/3

We know that 2 lines are perpendicular if the product of their slopes is -1.

The given equation *y *= -3*x *+ 7 is written in the standard from, which is y = mx+n, so the slope can be easily determined as m = -3.

The product of the slopes being -1, that means that:

-3*m1 = -1

We'll divide by -3 both sides:

m1 = 1/3

The line which passes through the point (-1, 5) and has the slope m1 = 1/3, has the equation:

y-5 = (1/3)(x+1)

We'll open the brackets:

y-5 = x/3 + 1/3

We'll add 5 both sides:

y = x/3 + 1/3 + 5

We'll add the terms 1/3 and 5, after multiplying 5 by 3:

**y = x/3 + 16/3**

It is obvious that the first given equation is the equation of the line which is perpendicular to *y *= -3*x *+ 7 and it passes through the point (-1, 5).