Can Someone Help me with this Calc problem

Question

The point P( 1/2, − 2/7 ) lies on the graph of y = 2x/x−4. If Q denotes the point (x, 2x/ x−4 ), find the simplified form of the slope of the line joining P and Q when x = 1/3

Answer 1

Answer:

We know Slope of line = change in y/ change in x

= (y₂ - y₁)/(x₂ - x₁)

Here, x₁ = 1/2, x₂ = x

and y₁ = -2/7, y₂ = 2x/(x - 4)

Now given that x = 1/3

Using values of x we get,

x₂ = 1/3 and y₂ = 2×(1/3)/(1/3 - 4)

= (2/3)/((1 - 12)/3)

= 2/3 × -3/11

= -2/11

Now slope = (y₂ - y₁)/(x₂ - x₁)

= (-2/11 - (-2/7))/(1/3-1/2)

= (-2/11 + 2/7)/(1/3-1/2)

= [(-14 + 22)/77]/[(2 - 3)/6]

= (8/77)/(-1/6)

= 8/77 × (-6/1)

= (8 × (-6))/77×1

= -48/77 is required answer.

Answer 2

At x = 1/3, Q = (1/3,–2/11).

The line connecting P and Q has simple equation

y = P₂ + ((Q₂ – P₂)/(Q₁ – P₁))(x – P₁)

= –2/7 + ((–2/11 + 2/7)/(1/3 – 1/2))(x – 1/2)

= –2/7 + ((8/77)/(–1/6))(x – 1/2)

= –2/7 – (48/77)(x – 1/2)

= 2/77 – (48/77)x.

I had made a mistake last night but now it's correct.

Answer 3

Q(x, 2x/(x-4))= (1/3, -2/11)

P(1/2, -2/7)

slope = "rise over run" = difference in y divided by difference in x =

(-2/7 +2/11)/(1/2- 1/3)=(y+2/7)/(x-1/2)

(-8/77)/(1/6)=-48/77=(y+2/7)/(x-1/2)

y=-48x/77 +24/77-2/7

y=-48x/77 +2/77 is the line connecting P and Q, with slope = -48/77