Entomology is the scientific study of insects. Patty, Ronny and Sammy collected some insects for their science project. At the end of the first week of collecting, Patty and Ronny together had 89 insects, Ronny and Sammy together had 80 insects, and Patty and Sammy together had85 insects. How many insects did each person have alone.

I get the main idea, but i just don't know how to write it out.
Thanks!!

P + R = 89

R + S = 80
P + S = 85

Now let's subtract equation 3 from equation 2
R - P = -5
now add that ot equation 1
P + R = 89
R - P = -5
------------
2 R = 84
so
R = 42
Now I will leave it to you to work your way back up and find P and S

in the first step, what happened to the S? I understand how you got -5, but I don't understand what happened to S. i understand everything else, just that.

would S = -5?

To solve this problem, we can use a system of equations. Let's assign variables to represent the number of insects each person had.

Let's say Patty had "P" insects, Ronny had "R" insects, and Sammy had "S" insects.

We can now write the following equations based on the information given:

1. Patty and Ronny together had 89 insects: P + R = 89

2. Ronny and Sammy together had 80 insects: R + S = 80

3. Patty and Sammy together had 85 insects: P + S = 85

Now we need to solve this system of equations to find the values of P, R, and S.

There are multiple ways to solve this system, such as substitution or elimination. Let's use the elimination method.

First, let's subtract equation 2 from equation 1 to eliminate R:

(P + R) - (R + S) = 89 - 80

P - S = 9 (Equation 4)

Now, let's subtract equation 3 from equation 1 to eliminate S:

(P + R) - (P + S) = 89 - 85

R - S = 4 (Equation 5)

Now we have a new system of equations with variables P and S.

Let's add equation 4 and equation 5 to eliminate S:

(P - S) + (R - S) = 9 + 4

P + R - 2S = 13 (Equation 6)

Now we have a new equation with P and R. We can substitute this equation into equation 3 to solve for P:

(P + S) = 85

P + 85 - P = 85

S = 0

Since S = 0, we can substitute this value into equation 4:

P - 0 = 9

P = 9

Now that we have P = 9, we can substitute this value into equation 1 to solve for R:

9 + R = 89

R = 80

So, Patty had 9 insects, Ronny had 80 insects, and Sammy had 0 insects.