The vet told jake that his dog, rocco, who weighed 55 pounds, needed to lose 10 pounds. jake started walking rocco every day and changed the amount of food he was feeding him. rocco lost half a pound the first week. jake wants to determine rocco’s weight in pounds, p, after w weeks if rocco continues to lose weight based on his vet’s advice.

Given:
Rocco's Weight = 55 pounds
Need to lose 10 pounds.
1 week = loses 1/2 pounds

It is safe to assume that Rocco is overweight by 10 pounds and his ideal Weight is 45 pounds

So, we need to find the number of weeks Rocco need to reach his ideal weight.

1/2 x = 45
x = 45 ÷ 1/2
x = 45 * 2/1
x = 90 weeks

Rocco needs to continue his rate of walking and food intake for at least 90 weeks to reach his ideal weight of 45 pounds.

If we use 55 pounds as basis,

1/2 x + 10 = 55
1/2 x = 55 - 10
1/2 x = 45
x= 45 ÷ 1/2
x = 45 * 2/1
x = 90/1 or 90 weeks

-- Following the vet's instructions, Rocco lost 1/2 pound in the
first week.  Jake is going to continue following the vet's instructions,
so Rocco will continue to lose 1/2 pound every week.  If you draw a
graph of Rocco's weight, its slope will be  (-1/2 pound per week).

-- When this whole thing started, Rocco weighed 55 pounds.
So when you draw the graph, the x-axis begins with the visit
to the vet, and the y-intercept of the graph is  55 pounds.

--  There's your equation for the graph. 

                       slope = -1/2 pound per week
                  intercept = 55 pounds

                      pounds  =  -1/2 weeks + 55

                               P  =  -1/2 w + 55

According to the vet, Rocco should weigh 45 pounds.
How many weeks does Jake need to follow the vet's instructions ?
Use your equation.  Make 'P'=45  and solve for 'w'.

                                                45  =  -1/2 w + 55

Subtract 45 from each side:      0  =  -1/2 w + 10

Add  1/2 w  to each side:    1/2 w  =                10

Multiply each side by 2 :           w  =                20

Rocco will weigh 45 pounds at the end of 20 weeks.

But if I may suggest ... walking every day and eating a
healthy amount of nutritious food will be a good thing for
Rocco to continue even after he reaches his target weight.
And it won't hurt Jake either.