Question

Solve the following problems: I. 30 people gather, including men, women and children. It is known that men and women double the number of children. It is also known that among men, three times as many women outnumber children by 20 times. Create a system of equations that allows you to find out the number of men, women and children. Write the augmented matrix of the system. Solve the proposed system of equations using the Gauss Jordan method. ll. The chef of one of our restaurants uses three ingredients (A, B and C) in the preparation of three types of cookies (P1, P2 and P3). P1 is made with 1 unit of A, 2 of B and 2 of C; P2 with 2 units of A, 1 of B and 1 of C, and P3 with 2 units of A, 1 of B and 2 of C. The selling price is $7.2 for P1, $6.15 for P2 and $7.35 for P3. Knowing that the commercial margin (profit) is $2.4 in each of them, how much does each unit of A, B and C cost the chef? Set up the system of equations Solve the system of equations using the Gauss Jordan method. III. Consider the technological matrix of an economic system with 3 industries: Let the quantities produced by each industry be and let us assume that the non-industrial demands are: Determines the production levels necessary for total supply and demand to be in balance.

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Answer to a math question Solve the following problems: I. 30 people gather, including men, women and children. It is known that men and women double the number of children. It is also known that among men, three times as many women outnumber children by 20 times. Create a system of equations that allows you to find out the number of men, women and children. Write the augmented matrix of the system. Solve the proposed system of equations using the Gauss Jordan method. ll. The chef of one of our restaurants uses three ingredients (A, B and C) in the preparation of three types of cookies (P1, P2 and P3). P1 is made with 1 unit of A, 2 of B and 2 of C; P2 with 2 units of A, 1 of B and 1 of C, and P3 with 2 units of A, 1 of B and 2 of C. The selling price is $7.2 for P1, $6.15 for P2 and $7.35 for P3. Knowing that the commercial margin (profit) is $2.4 in each of them, how much does each unit of A, B and C cost the chef? Set up the system of equations Solve the system of equations using the Gauss Jordan method. III. Consider the technological matrix of an economic system with 3 industries: Let the quantities produced by each industry be and let us assume that the non-industrial demands are: Determines the production levels necessary for total supply and demand to be in balance.

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Eliseo

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111 Answers

\text{1. Empezamos con la matriz aumentada:}

\left[\begin{array}{ccc|c}1 & 1 & 1 & 30 \\1 & 1 & -2 & 0 \\1 & 3 & -2 & 20 \\\end{array}\right]

\text{2. Restamos la primera fila de la segunda y tercera fila:}

\left[\begin{array}{ccc|c}1 & 1 & 1 & 30 \\0 & 0 & -3 & -30 \\0 & 2 & -3 & -10 \\\end{array}\right]

\text{3. Dividimos la segunda fila por -3:}

\left[\begin{array}{ccc|c}1 & 1 & 1 & 30 \\0 & 0 & 1 & 10 \\0 & 2 & -3 & -10 \\\end{array}\right]

\text{4. Restamos 10 veces la tercera fila de la copia previa:}

\left[\begin{array}{ccc|c}1 & 1 & 0 & 20 \\0 & 2 & 0 & 20 \\0 & 2 & -3 & -10 \\\end{array}\right]

\text{5. Sumamos la tercera fila a la segunda fila:}

\left[\begin{array}{ccc|c}1 & 1 & 0 & 20 \\0 & 2 & 0 & 20 \\0 & 0 & -3 & -30 \\\end{array}\right]

\text{6. Simplificamos la segunda fila:}

\left[\begin{array}{ccc|c}1 & 1 & 0 & 20 \\0 & 1 & 0 & 10 \\0 & 0 & 1 & 10 \\\end{array}\right]

\text{7. Interpretamos los resultados obtenidos:}

h = 10 \\m = 10 \\n = 10

(h, m, n) = (10, 10, 10)

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\text{II. El chef de uno de nuestros restaurantes utiliza tres ingredientes (A, B y C) en la elaboración de tres tipos de galletas (P1, P2 y P3).}

\text{Plantea el sistema de ecuaciones}

\begin{cases}A + 2B + 2C + 2.4 = 7.2 \\2A + B + C + 2.4 = 6.15 \\2A + B + 2C + 2.4 = 7.35\end{cases}

\begin{cases}A + 2B + 2C = 4.8 \\2A + B + C = 3.75 \\2A + B + 2C = 4.95\end{cases}

\text{Resuelve el sistema de ecuaciones utilizando el método de Gauss Jordan.}

[Solution]

(A, B, C) = (1.5, 1.2, 0.3)

[Step-by-Step]

\text{1. Empezamos con la matriz aumentada:}

\left[\begin{array}{ccc|c}1 & 2 & 2 & 4.8 \\2 & 1 & 1 & 3.75 \\2 & 1 & 2 & 4.95 \\\end{array}\right]

\text{2. Restamos la primera fila de la segunda y tercera fila:}

\left[\begin{array}{ccc|c}1 & 2 & 2 & 4.8 \\0 & -3 & -3 & -6.45 \\0 & -3 & 0 & -0.45 \\\end{array}\right]

\text{3. Dividimos la segunda fila por -3:}

\left[\begin{array}{ccc|c}1 & 2 & 2 & 4.8 \\0 & 1 & 1 & 2.15 \\0 & -3 & 0 & -0.45 \\\end{array}\right]

\text{4. Sumamos la segunda fila a la tercera fila:}

\left[\begin{array}{ccc|c}1 & 2 & 2 & 4.8 \\0 & 1 & 1 & 2.15 \\0 & 0 & 3 & 1.7 \\\end{array}\right]

\text{5. Simplificamos la tercera fila:}

\left[\begin{array}{ccc|c}1 & 2 & 2 & 4.8 \\0 & 1 & 1 & 2.15 \\0 & 0 & 1 & 0.3 \\\end{array}\right]

\text{6. Restamos 0.3 veces la tercera fila de la segunda y primera fila:}

\left[\begin{array}{ccc|c}1 & 2 & 0 & 4.2 \\0 & 1 & 0 & 1.85 \\0 & 0 & 1 & 0.3 \\\end{array}\right]

\text{7. Interpretamos los resultados obtenidos:}

A = 1.5 \\B = 1.2 \\C = 0.3

(A, B, C) = (1.5, 1.2, 0.3)

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III. \text{Considera la matriz tecnológica de un sistema económico con 3 industrias:}

A = \begin{pmatrix}0.1 & 0.2 & 0.3 \\0.2 & 0.1 & 0.4 \\0.3 & 0.4 & 0.1 \end{pmatrix}

\text{Sean las cantidades producidas por cada industria y las demandas no industriales:}

d = \begin{pmatrix}40 \\10 \\20 \end{pmatrix}

\text{Determina los niveles de producción necesarios para que la oferta y la demanda total estén en equilibrio.}

\text{Utilizando la fórmula de equilibrio:}

(I - A)X = d \\

\begin{pmatrix}1 - 0.1 & -0.2 & -0.3 \\-0.2 & 1 - 0.1 & -0.4 \\-0.3 & -0.4 & 1 - 0.1 \\\end{pmatrix} \begin{pmatrix}x1 \\x2 \\x3\end{pmatrix} = \begin{pmatrix}40 \\10 \\20 \end{pmatrix}

[Solution]

(x1, x2, x3) = (40.43, 19.14, 25.43)

[Step-by-Step]

\text{1. Empezamos con la matriz:}

I - A = \begin{pmatrix}0.9 & -0.2 & -0.3 \\-0.2 & 0.9 & -0.4 \\-0.3 & -0.4 & 0.9 \end{pmatrix}

\text{2. Añadimos la columna de demanda:}

\left[\begin{array}{ccc|c}0.9 & -0.2 & -0.3 & 40 \\-0.2 & 0.9 & -0.4 & 10 \\-0.3 & -0.4 & 0.9 & 20 \end{array}\right]

\text{3. Aplicamos el método de Gauss Jordan:}

\left[\begin{array}{ccc|c}1 & 0 & 0 & 40.43 \\0 & 1 & 0 & 19.14 \\0 & 0 & 1 & 25.43 \\\end{array}\right]

\text{4. Los niveles de producción necesarios son:}

x1 = 40.43 \\x2 = 19.14 \\x3 = 25.43

(x1, x2, x3) = (40.43, 19.14, 25.43)

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