Applications of linear systems in two unknowns with a tabular setup

In many situations, a real world situation lends itself to a system of linear equations with a tabular setup

#[Here is a typical (generic) situation of this type:][He aquí una situación típica (genérica) de este tipo:]#

#[Unknowns][Desconocidos]#: #[To find the unknowns, we look at the question part: "How many of each product...?" This suggests that the unknowns are the number of each product type. So we take][Para hallar las incógnitas, nos fijamos en la pregunta: "¿Cuántos de cada producto...?". Esto sugiere que las incógnitas son la cantidad de cada tipo de producto. Así que tomamos]#

$x = $ #[Number of units of Product A][Número de unidades del Producto A]# \\ $y = $ #[Number of units of Product B][Número de unidades del Producto B]# \\ $z = $ #[Number of units of Product C][Número de unidades del Producto C]#

We now arrange the given information in a table with the categories represented by the unknowns across the top as follows:

#[We get the following system of equations by simply reading the table as we have set it up to find the coefficients.][Obtenemos el siguiente sistema de ecuaciones simplemente leyendo la tabla tal como la hemos configurado para encontrar los coeficientes.]#

$2x + 5y + 7z = 45$ \\ $4x + 3y + 2z = 26$ \\ $6x + 3y + 4z = 36$

#[We can then solve it using any of the online tools (like the %1) and obtain the solution: $x = 2$ units of product A and $y = 4$ units of product B.][Luego podemos resolverlo usando cualquiera de las herramientas en línea (como %1) y obtener la solución: $x = 2$ unidades del producto A y $y = 4$ unidades del producto B.]#

One for you:

The Scrumptious %30 company makes three brands of %30 for vendors: %31, using %21 of %35, %22 of %36, and %23 of %37 per %34, %32, using %24 of %35, %25 of %36, and %26 of %37 per %34, and %33, using %27 of %35, %28 of %36, and %29 of %37 per %34. Each day the company has %45 %38s of %35, %46 %38s of %36, and %47 %38s of %37 available. How many %34s of each brand should it make each day if it wants to use up all of the supplies?

#[Step][Paso]# 1. #[Specify the unknowns in this situation. (Hint: What is the question asking for?)][Especifica las incógnitas en esta situación. (Pista: ¿Qué pregunta la pregunta?)]# #[The unknowns are][Las incógnitas son]#

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#[Step][Paso]# 2. #[In this kind of application we can tabulate all the given information with the categories corresponding to the unknowns along the top. Which of the following is the resulting table layout?][En este tipo de aplicación podemos tabular toda la información dada con las categorías correspondientes a las %unknowns a lo largo de la parte superior. ¿Cuál de las siguientes el diseño tabular resultante?]#
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#[Step][Paso]# 4. #[Use the table to set up a system of linear equations.][Utiliza la tabla para configurar un sistema de ecuaciones lineales.]#

#[Using the order suggested by the configuration of the table, we obtain the following system of equations:][Utilizando el orden sugerido por la configuración de la tabla, obtenemos el siguiente sistema de ecuaciones:]#

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#[Step][Paso]# 5.#[Solve the systems in Step 4.][Soluciona el sistema en Paso 4.]#

In order to exactly use all the available resources, the company should make BOX %34 of %31 %30, BOX %34 of %32 %30 and BOX %34 of %33 %30.

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