Question 1:Solve this system of equations: y = 2x², y = -3x – 1
We can only solve equations with one variable.
Since both right sides of each equation is equal to y, we can set them equal to each other.
2x² = -3x – 1
Let’s get everything on the left side…
2x² + 3x + 1 = 0
Let’s solve this quadratic by splitting the middle…
We want two #’s that add up to 3 and multiply to 2. (2×1)
These are 2 and 1. Let’s split the middle into 2x and x.
2x² + 2x + x + 1 = 0
Factor the first and last two terms.
2x(x+1)+1(x+1) = 0
(2x+1)(x+1) = 0
If any value of x makes either factor evaluate to 0, it is a solution.
2x + 1 = 0 ⇒ 2x = -1 ⇒ x = -1/2 ⇒ y = 2(-1/2)² = 2(1/4) = 1/2
x + 1 = 0 ⇒ x = -1 ⇒ y = 2(-1)² = 2(1) = 2
Our solutions are (-1/2, 1/2) and (-1, 2).
Of course, if you haven’t been taught how to solve quadratics yet, you could just use guess and check. Try a bunch of different values for x and see if you end up with the same y value.

Question 2:

The Mathalot Company makes and sells textbooks. They have one linear function that represents the cost of producing textbooks and another linear function that models how much income they get from those textbooks. Describe the key features that would determine if these linear functions ever intercepted.
If two lines have the same slope, they are parallel and will never intercept. If they don’t have the same slope, they will intercept.

Question 3Your boss hands you the monthly data that show the number of orders coming in to and out of the warehouse. The data are in the table below. Explain to your boss, in complete sentences, the solution to this system and what the solution represents.
If the orders coming in and out were the same at any point, then the warehouse would be empty. (Data: month/in/out–jan/3/3, feb/6/4, mar/9/5, apr/12/6) The obvious intersection here is in January when the orders in and orders out are both 3.