(06.01)
solve the following system of equations and show all work.
y = 2x2
y = –3x −1
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.
your 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.
month
no. of orders in
no. of orders out
january (1)
3 3
february (2)
6 4
march (3)
9 5
april (4)
12 6
in any of them

These are three questions and three answers.

Question 1: Solve the following system of equations:

y = 2x², y = -3x - 1

i) (-1, 2)

ii) (-1/2, 1/2)

Explanation:

1) Make y = y:

2x² = - 3x - 1

2) Transpose all the terms to the left side:

2x² + 3x + 1 = 0

3) Regroup for factoring:

2x² + 2x + x + 1 = 0 ← slplit 3x into 2x + 1

[2x² + 2x] + [x + 1] = 0 ← group

2x (x + 1) + (x + 1) = 0 ← common factor 2x in the first group

(x + 1) (2x + 1) = 0 ← common factor x + 1

3) Set each factor equal to 0 and solve:

i) x + 1 = 0 ⇒ x = - 1

ii) 2x + 1 = 0 ⇒ x = - 1/2

4) Find y for each solution, using any of the initial equations:

i) y = - 3x -1 = - 3(-1) - 1 = 2

ii) y = -3(-1/2) - 1 =1/2

5) Solutions:

i) (-1, 2)

ii) (-1/2, 1/2)

Question 2: The Mathalot Company makes and sells textbooks...

1) Two variable lnear functions are represented by lines in the plane.

2) If a system of two variables linear functions have solutions the lines intercept in the plane.

3) The interception represents the solution.

4) Two lines intercept and have one solution if they are different non-parallel lines.

This is, the two lines must have different slopes. If they have equal slope, they either are parallel, and never intercept (the system do not have solution), or they are the same line and there are infinite solutions (you cannot idenfity one single solution).

Those are the key features that will determine whether the system has one solution.

On the other hand, you also should know what the solution means. It is the number of books for which the income and the cost are equal, and such value.

Question 3: orders coming in to and out of the warehouse.

1) This is the table of data:

Month No. of orders in No. of orders out

January (1) 3 3

February (2) 6 4

March (3) 9 5

April (4) 12 6

2) Explanation:

The system is formed by the number of orders in and the number of order out on four months (January through April).

The solution is when the number of orders in and the number of orders out are the same.

That only happens in January and the number of orders is 3.

So, the solution is 3 orders in and out, in January.

You can predict that, if the same conditions remain in the future, the number of orders in and out will not be equal again, but since the domain of the table is only four months, there is no way to tell that. The only conclusion, that you can infere is that the the solution of the sysment is the month of April with 3 books.

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step-by-step explanation:

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