I posted this yesterday and no one bothered to check it, therefore Im posting it again, please check it this time.

The south edmonton pet shop has several parrots and dogs for sale. There are a total of 24 heads and 82 legs in the display cages.

A) Write a system of linear equations to represent the number of parrots, p, and the number of dogs, d, for sale.

p+d = 24
2p+4d = 82

B) Determine the solution to this system graphically(done it)

c)Explain why this system of linear equations would have no solution if the total number of legs is changed from 82 to 83?

Cause it wouldnt equal up with eachother? no sure how to rephrase this into a better answer.

D) Why is your answer to part C not related to the slopes of the two lines?

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c) the solution for both p and d must be a positive integer.

Look at your second equation.
The left side is an even number, (anything multiplied by either 2 or 4 is even)
so you would have an even number equal to an odd number, which cannot be.
If you try to solve the system with the 2nd equation equal to 83, you would end up with a fraction as a solution. How can you have a fraction of a parrot or a fraction of a dog?

d) changing the 82 to 83 does not change the slope of the line, it simply raises the line parallel to itself.
The "nice" intersection point consisting of whole numbers now moves up as well, "wrecking" our nice numbers.

A)

Equation 1: 2p + 4d = 82
Equation 2: p + d = 24

Now you have to isolate a pronumeral from equation 2. For example:

Equation 3: d = 24 - p

Now you know what the value of 'd' is, you then substitute that into equation 1.

Which will equal:
2p + 4(24-p) = 82

2p + 96 - 4p = 82

-2p = -14

p = 7

Now that you know how many Parrots there were you simply just substitute into any of the equations.

Therefore d = 17 & p = 7

Note: if you have done simultaneous equations it would make more sense.

Hope this helps! :)

To answer part C, let's consider the equations in the system:

p + d = 24 (Equation 1)
2p + 4d = 82 (Equation 2)

In this system, Equation 1 represents the total number of animals (parrots and dogs) being equal to 24. Equation 2 represents the total number of legs of the animals being equal to 82.

If the total number of legs is changed from 82 to 83, there would be no solution to this system of equations. This is because the system represents a situation where each animal (parrot or dog) contributes a certain number of heads and legs. Parrots have one head and two legs, while dogs have one head and four legs.

When the total number of legs is 82, every animal has the correct number of legs based on their type (parrot or dog). However, if the total number of legs is changed to 83, it would imply that there is a fractional number of animals or a new type of animal with a different number of legs. Since the variables in the system are restricted to whole numbers representing the number of animals, there is no way to have a fractional number of animals in the system. Hence, there would be no solution.

Moving on to part D, the answer is not related to the slopes of the two lines because the slopes represent the rate of change or the ratio of the change in one variable to the change in the other variable. In this case, the slopes of the lines formed by the equations are not influential in determining if a solution exists or not. The lack of solution in part C is due to the fact that the system represents a real-world situation where the number of animals and their legs must align, rather than being determined solely based on the slopes of the equations.