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.