Question

A business has 40,000 to spend on advertising in an upcoming sale. The money is to be divided between television (x), radio (y) and newspapers (z). The business manager has decided to spend three times as much money on television as on radio. The manager has also decided to spend 8000 less on radio advertising than on newspapers. Find out, using matrix inversion method, the total amount divided between television, radio and newspapers.​

Answer 1

Let's start by setting up a system of equations to represent the given information.

From the problem, we know that:

x + y + z = 40000 (the total amount of money to be spent is 40,000)

x = 3y (three times as much money is being spent on television as on radio)

y = z - 8000 (8000 less is being spent on radio advertising than on newspapers)

We can use this system of equations to set up a matrix equation:

```
[1 1 1] [x] [40000]
[1 -3 0] * [y] = [0]
[0 1 -1] [z] [8000]
```

To solve for x, y, and z, we need to invert the coefficient matrix and multiply it by the right-hand side vector:

```
[1 1 1] [1/4 1/4 -1/4] [40000] [15000]
[1 -3 0] * [-1/12 -1/12 1/4] * [0] = [0]
[0 1 -1] [1/12 1/12 1/4] [8000] [12500]
```

Therefore, the total amount of money divided between television, radio, and newspapers is:

x = 15000

y = 0

z = 12500

This means that the business should spend $15,000 on television advertising, $0 on radio advertising, and $12,500 on newspaper advertising.

Answer 2

Okay, let's break this down step-by-step:

* The business has $40,000 total to spend on advertising

* Some will go to TV (x), some to radio (y), and some to newspapers (z)

* 3 times as much will go to TV as radio, so x = 3y

* They will spend $8,000 less on radio than newspapers, so y = z - 8,000

* We have:

x = 3y (1)

y = z - 8,000 (2)

x + y + z = 40,000 (3)

To solve this using matrix inversion:

1) Turn the equations into a matrix:

3 1 0 y

1 -1 1 z

1 1 1 x

2 0 40,000

2) Invert the matrix:

0.3333 0.3333 0.3333

-0.25 0.75 0

0.125 0.125 0.750

3) Plug in the values from (2) and (3):

y = 0.3333(z - 8,000)

x = 0.125z + 0.125(40,000 - z)

4) Solve for z, the amount for newspapers. We get:

z = 40,000 * (0.75) = 30,000

5) Plug z = 30,000 back into the other equations:

x = 0.125 * 30,000 + 0.125 * 10,000 = 12,000

y = 0.3333 * (30,000 - 8,000) = 8,000

z = 30,000

So in total:

TV (x) = $12,000

Radio (y) = $8,000

Newspapers (z) = $30,000

Does this make sense? Let me know if you have any other questions!