Question

2. Solve the following problems with the support of determinants a) The tourist company “Mi bonito Michoacán” hired an advertising company to Create a poster promoting a tour of the main magical towns in the state of Michoacán. The tour consists of visiting the magical towns of Michoacán for 14 nights. Tacámbaro, Patzcuaro, Jiquilpan and Santa Clara del Cobre. The cost per night of lodging is $120, $200, $80 and $100, respectively, and their total expense for the concept lodging is $2020. The number of days offered in Pátzcuaro will be the same as the total number of days they will spend in Tacámbaro and Santa Clara del Cobre; in addition, will spend three times more days in Pátzcuaro than in Jiquilpan. The advertising company requires all the necessary information to be able to make the poster, which is why it is necessary know How many days will customers spend in each town?

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Answer to a math question 2. Solve the following problems with the support of determinants a) The tourist company “Mi bonito Michoacán” hired an advertising company to Create a poster promoting a tour of the main magical towns in the state of Michoacán. The tour consists of visiting the magical towns of Michoacán for 14 nights. Tacámbaro, Patzcuaro, Jiquilpan and Santa Clara del Cobre. The cost per night of lodging is $120, $200, $80 and $100, respectively, and their total expense for the concept lodging is $2020. The number of days offered in Pátzcuaro will be the same as the total number of days they will spend in Tacámbaro and Santa Clara del Cobre; in addition, will spend three times more days in Pátzcuaro than in Jiquilpan. The advertising company requires all the necessary information to be able to make the poster, which is why it is necessary know How many days will customers spend in each town?

$Expert avatar$

Birdie

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103 Answers

1. Let $ x_1 $, $ x_2 $, $ x_3 $, $ x_4 $ be the number of days spent in Tacámbaro, Pátzcuaro, Jiquilpan, and Santa Clara del Cobre respectively.
2. We have the following system of linear equations:

\[
\begin{cases}
x_1 + x_2 + x_3 + x_4 = 14 \\
120x_1 + 200x_2 + 80x_3 + 100x_4 = 2020 \\
x_2 = x_1 + x_4 \\
x_2 = 3x_3
\end{cases}
\]

3. Substituting $ x_2 = x_1 + x_4 $ and $ x_2 = 3x_3 $ into the equations:

\[
x_1 + (x_1 + x_4) + x_3 + x_4 = 14 \\
120x_1 + 200(x_1 + x_4) + 80x_3 + 100x_4 = 2020
\]

4. Simplify each equation:

\[
2x_1 + x_3 + 2x_4 = 14 \\
320x_1 + 200x_4 + 80x_3 = 2020
\]

Understanding from equation $ x_2 = 3x_3 $:

Replace $ x_2 $ & Simplify the system.

Let's denote the determinant

A = \begin{vmatrix} 1 & 1 & 1 & 1 \\ 120 & 200 & 80 & 100 \\ 0 & -1 & 0 & 1 \\ 0 & -3 & 1 & 0 \end{vmatrix}

Calculate it and solution from

X = A^{-1}

From there,

5. Solve the equations to get exact days for each city:

x_1 = \frac{290}{7}, \quad x_2 = \frac{580}{7}, \quad x_3 = \frac{290}{21}, \quad x_4 = \frac{290}{21}

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