Write a system of linear equations in three or four...

Write a system of linear equations in three or four variables to solve Exercises $47-50 .$ Then use matrices to solve the system. A furniture company produces three types of desks: a children's model, an office model, and a deluxe model. Each desk is manufactured in three stages: cutting, construction, and finishing. The time requirements for each model and manufacturing stage are given in the following table. $$ \begin{array}{lccc} \hline & \begin{array}{c} \text { Children's } \\ \text { Model } \end{array} & \begin{array}{c} \text { Office } \\ \text { Model } \end{array} & \begin{array}{c} \text { Deluxe } \\ \text { Model } \end{array} \\ \hline \text { Cutting } & 2 \mathrm{hr} & 3 \mathrm{hr} & 2 \mathrm{hr} \\ \text { Construction } & 2 \mathrm{hr} & 1 \mathrm{hr} & 3 \mathrm{hr} \\ \text { Finishing } & 1 \mathrm{hr} & 1 \mathrm{hr} & 2 \mathrm{hr} \end{array} $$ Each week the company has available a maximum of 100 hours for cutting, 100 hours for construction. and 65 hours for finishing. If all available time must be used, how many of each type of desk should be produced each week?

Write a system of linear equations in three or four variables to solve Exercises $47-50 .$ Then use matrices to solve the system.
A furniture company produces three types of desks:
a children's model, an office model, and a deluxe model. Each desk is manufactured in three stages: cutting, construction, and finishing. The time requirements for each model and manufacturing stage are given in the following table.
$$
\begin{array}{lccc}
\hline & \begin{array}{c}
\text { Children's } \\
\text { Model }
\end{array} & \begin{array}{c}
\text { Office } \\
\text { Model }
\end{array} & \begin{array}{c}
\text { Deluxe } \\
\text { Model }
\end{array} \\
\hline \text { Cutting } & 2 \mathrm{hr} & 3 \mathrm{hr} & 2 \mathrm{hr} \\
\text { Construction } & 2 \mathrm{hr} & 1 \mathrm{hr} & 3 \mathrm{hr} \\
\text { Finishing } & 1 \mathrm{hr} & 1 \mathrm{hr} & 2 \mathrm{hr}
\end{array}
$$
Each week the company has available a maximum of 100 hours for cutting, 100 hours for construction. and 65 hours for finishing. If all available time must be used, how many of each type of desk should be produced each week?

Key Concepts

System of Linear Equations

A system of linear equations consists of multiple equations with several variables that are related linearly. In this context, each equation represents a constraint—such as resource usage or production requirements—and the variables typically denote the quantities to be determined. The solution is the set of variable values that simultaneously satisfies all the equations in the system.

Matrix Representation

Matrices provide a concise and structured way to represent systems of linear equations. By arranging the coefficients of the variables into a matrix, along with corresponding constant terms in a vector, one can streamline the process of applying systematic methods to solve the system, such as elimination or inversion techniques.

Gaussian Elimination

Gaussian elimination is a method used to solve systems of linear equations by transforming the matrix into a row-echelon form. This process involves performing a series of row operations so that the system is simplified step-by-step, ultimately allowing for back substitution to find the values of the variables.

Modeling Resource Constraints

In real-world applications, such as production planning, resource constraints are modeled using linear equations to represent the limitations or capacities (like time, material, or manpower) available. Each equation corresponds to a different resource, and the coefficients in the equations represent the amount of resource consumed by each unit of production, linking the abstract mathematical system to practical decision-making problems.

Transcript

00:08 Write a system of linear equations and three variables and use matrices to solve the system.

00:14 A furniture company produces three types of desks, children's model, an office model, and a deluxe model.

00:20 Each desk is manufactured in three stages, cutting, construction, and finishing.

00:24 The time requirements for each model and manufacturing stage are given in a falling table.

00:29 Each week the company has available a maximum of 100 hours of cutting, 100 hours of construction, and 65 hours of finishing.

00:36 If all the available time of it must be used, how many of each type of desks should be produced each week.

00:42 And so you have the table there, and we need to figure out how many of each model we're going to produce.

00:50 And so, for instance, a children's model for cutting, it takes two hours times however many desks we're going to produce.

01:00 And so if we want all of the three together to add up to be 100, then we're going to need to take an add -up.

01:08 Up how many children's model desks times two hours plus office models times three plus the deluxe model times two and then when we add that together it should equal a hundred but since we don't know how many we are making of each desk we're going to use a variable and so we're going to start out with for cutting we're going to have two and we're going to let the children's model be x the office b y and the deluxe b z for the number produced of each and so we're going to take, say, two hours times x number of desks, plus, that'd be the x children's wild desk, and then plus 3y plus 2z is going to equal 100.

01:59 And then we're going to do the same thing for the construction.

02:03 We want 2x plus y plus 3z to equal 100, and then the same thing for the finishing.

02:14 We want x, 1x, in other words, plus 1y, plus 2z to equal 65.

02:26 And then we're going to take and write those, oops, not too far, into an augmented matrix so that we can use the matrices to solve.

02:35 So, our first line is going to be 2, 3, 2, and 100.

02:41 Second line will be 2, 1, 3, 100.

02:46 Third line will be 1, 1, 2, and 65.

02:54 So step 1, we are going to take and switch row 1 and row 3 because we want to get ones on the diagonal, so we need this first one to hear to be 1.

03:04 Since we have a 1 in the 3rd already, we're going to just take row 1 and switch it with row 3.

03:10 And so we're going to have 1 ,1, whoops, moon is blue.

03:22 So we're going to have 1, 1, 2, and 65.

03:28 And then we're going to have two, one, three, and a hundred.

03:32 And then we're putting row one where row three was.

03:38 Okay, so now we have one right here.

03:41 We need everything underneath that to be a zero.

03:45 And so we're going to take and multiply row one.

03:52 So row two, we're going to take row two and replace it with row one times the negative two.

04:01 And then we're going to add, so that would be row one.

04:04 We're going to add row two to it.

04:05 And so row 1 times a negative 2 would be negative 2, negative 2, negative 2, negative 4, and then 65 times 2 would be 130, so negative 130.

04:28 And so row 1, we're going to leave the same.

04:33 Row 2, we're trying to get that to be a 0, so we're going to take and add to it twice row 1.

04:40 So we need to have row 2 here, which would be 2, 1, 3, and 100.

04:48 And then we're going to add these together to get 0, negative 1, negative 1, and negative 30.

05:12 Okay.

05:18 Then we also want this to be a 0.

05:22 So we're going to take into replace row 3 with minus 2 times row 1 plus row 3.

05:35 And so again, our row 1 will be negative 2, negative 2, negative 4, and negative 130.

05:41 That's row 1 times negative 2.

05:44 And then row 3 is 2, 3, 2, and 100.

05:54 Looked up in that, 2 in there too, just to show that set group.

05:59 And then you have 2 plus 2 is 0.

06:01 Negative 2 plus 3 is 1.

06:03 Negative 4 plus 2 is a negative 2.

06:06 913 plus 100 is negative 30.

06:24 Now we need row 2.

06:34 This is needed to be a positive 1.

06:36 So essentially we're going to keep row 1 the same.

06:41 And then on row 2, if we just take it times the negative 1, it'll switch it.

06:49 So we want negative 1 times row 2.

06:55 That's going to give us 0, 1, 1, and negative 30.

06:59 And then we want this right here to be a 0.

07:09 Well, if we take and add these two rows together, then that'll become a zero.

07:15 So i'm going to say row two plus row three, but you've got to remember row two plus row three comes from up here.

07:26 Okay? and so you're going to have zero, and negative one plus one is zero, negative one plus a negative two be negative three, and then 30 plus a negative 30 would be zero.

07:45 Okay, now we need this to be a one.

07:50 So i'm going to keep all the rest of that the same for right now.

07:53 1 -1 -2 -65, 0 -1 and negative 30...

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