Give an example of a function using a set of at least 4 ordered pairs. The domain will be any four integers between 0 and +10. The range will be any four integers between -12 and 5. Explain why your example models a function. Give an example of at least four ordered pairs that does not model a function. The domain will be any four integers between 0 and +10. The range will be any four integers between -12 and +5. Explain why your example does not model a function.

2. Select any two integers between -12 and +12 which will become solutions to a system of two equations.
Write two equations that have your two integers as solutions. Show how you built the equations using your integers. Solve your system of equations by the addition/subtraction method. Make sure you show the necessary 5 steps.

Sure! Let's start with the first question:

1. Example of a function using 4 ordered pairs:
Let's choose the following ordered pairs for our example: (2, -10), (4, -5), (6, -1), (8, 2).

To explain why this example models a function, we need to ensure that each x-value in the domain is associated with a unique y-value in the range. In other words, there should not be any repeated x-values with different y-values.

In our example, each x-value (2, 4, 6, 8) is associated with a unique y-value (-10, -5, -1, 2). There are no repeated x-values, so each x-value has only one corresponding y-value. Therefore, this example models a function.

2. Example of ordered pairs that do not model a function:
Let's choose the following ordered pairs for our example: (2, -10), (4, -5), (4, -1), (8, 2).

In this example, we have a repeated x-value (4) with different y-values (-5, -1). This violates the definition of a function, where each x-value should have a unique y-value. Therefore, this example does not model a function.

Moving on to the second question:

2. Select any two integers between -12 and +12 which will become solutions to a system of two equations.
Let's choose the integers 4 and 6.

To build the equations, we need to use the chosen integers as the solutions. Let's say the first equation will involve the variable x and the second equation will involve the variable y.

Equation 1:
We can set x = 4, so the equation becomes 2x + y = 12.

Equation 2:
We can set y = 6, so the equation becomes 4x - 3y = 12.

Now, let's solve this system of equations using the addition/subtraction method:

Step 1: Multiply Equation 1 by 3:
6x + 3y = 36.

Step 2: Multiply Equation 2 by 2:
8x - 6y = 24.

Step 3: Add Equation 1 and Equation 2:
6x + 3y + 8x - 6y = 36 + 24,
14x - 3y = 60.

Step 4: Rearrange the equation to isolate the variable:
14x = 60 + 3y.

Step 5: Divide both sides by 14 to solve for x:
x = (60 + 3y)/14.

Now we have the value of x in terms of y. To find the specific solutions, we can substitute the chosen value of y (6) into the equation.

x = (60 + 3(6))/14 = (60 + 18)/14 = 78/14 = 39/7.

Therefore, the solution to the system of equations is x = 39/7 and y = 6.

I hope this explanation helps you understand the process of finding examples of functions and solving systems of equations! Let me know if you have any further questions.

Example of a function:

Let's say the domain is {1, 3, 5, 7} and the range is {-8, -4, 0, 3}. We can represent this function using the ordered pairs:

{(1, -8), (3, -4), (5, 0), (7, 3)}

This example models a function because each input value (domain) has a unique output value (range). In other words, for each x-value in the domain, there is only one y-value in the range. There are no repeated x-values with different y-values.

Example that does not model a function:

Let's say the domain is {2, 4, 6, 8} and the range is {-3, -8, -3, 2}. We can represent this set of ordered pairs as:

{(2, -3), (4, -8), (6, -3), (8, 2)}

This example does not model a function because the input value "6" in the domain has two different output values in the range (-3 and 2). In a function, each input value must have a unique output value. Since the ordered pair (6, -3) and (6, 2) both exist in the set, it violates the definition of a function.

Solving a system of equations:

Let's say the two integers we choose are 2 and 4. We will use these as the solutions for our system of equations.

Step 1: Write the equations
We can write two equations by substituting the chosen integers into variables. Let's say we have:
Equation 1: 3x + y = 10
Equation 2: 2x - y = -2

Step 2: Rearrange the equations
We can rearrange Equation 1 to solve for y: y = 10 - 3x.
We can rearrange Equation 2 to solve for y as well: y = 2x + 2.

Step 3: Set the two equations equal to each other:
10 - 3x = 2x + 2

Step 4: Solve for x
10 = 5x + 2
8 = 5x
x = 8/5

Step 5: Substitute x back into one of the original equations to solve for y
Using Equation 1: 3(8/5) + y = 10
24/5 + y = 10
y = 10 - 24/5
y = 26/5

Therefore, the solutions to the system of equations are x = 8/5 and y = 26/5.