Use the four-step strategy to solve each problem. Use x, y,...

Use the four-step strategy to solve each problem. Use $x, y,$ and $z$ to represent unknown quantities. Then translate from the verbal conditions of the problem to a system of three equations in three variables. 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. CAN'T COPY THE FIGURE 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?

Use the four-step strategy to solve each problem. Use $x, y,$ and $z$ to represent unknown quantities. Then translate from the verbal conditions of the problem to a system of three equations in three variables.
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.
CAN'T COPY THE FIGURE
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

Constraints Representation

Constraints in a problem often come from limited resources or conditions that must be met, such as time or capacity restrictions. In mathematical modeling, each resource or limitation can be represented by an equation that shows how the use of that resource is distributed among various components of the problem, ensuring the total doesn’t exceed the available limit.

Translating Verbal Conditions

Translating verbal conditions into mathematical equations involves reading the problem carefully, identifying the relationships and constraints described in words, and expressing these relationships as algebraic equations. This skill is essential because it bridges the gap between a real-world scenario and its mathematical representation, enabling the use of algebraic methods to find a solution.

Four-Step Problem Solving Strategy

The four-step strategy typically includes: (1) defining the variables to represent unknown quantities, (2) formulating the system of equations based on the conditions given in the problem, (3) solving the system of equations using an appropriate method, and (4) verifying the solution in the context of the problem. This structured approach facilitates a clear and organized way to tackle word problems effectively.

Variables Representation

In problems like this one, unknown quantities are represented using variables (such as x, y, and z). Representing unknowns with symbols allows for the formulation of equations that model the relationships described in the problem. This abstraction is essential in algebra for formulating and solving systems of equations.

System of Linear Equations

A system of linear equations comprises multiple linear equations that share the same set of unknown variables. Each equation represents a different condition or constraint from the problem. The solution to the system is the set of values for the variables that simultaneously satisfies all the equations, making this concept crucial in modeling multiple interrelated conditions in a problem scenario.

Transcript

00:01 In this problem, we have three different models of desks, model x, model y, and model z, and we have three steps in the desk making process.

00:11 So we have cutting, construction, and finishing.

00:14 And for each model, the steps take a different amount of time.

00:18 But we're given three total available amounts of time.

00:24 So we have 100 hours available for cutting, 100 for construction, and then 65 hours for finishing.

00:30 And so we want to figure out which type of desk should be produced each week if we have this amount of time each week for each step.

00:39 And so from this information, we can write a system of three equations to figure out how many of x, y, and z should be produced.

00:48 So our first equation is going to be the cutting equation.

00:51 So we have two hours for x for cutting x.

00:56 So if we take 2 times x, we will get the total amount of time that we need to cut the models of the x desk.

01:07 And then we can do the same thing for y, so 3 times y plus 2 times z.

01:13 And then the total amount of cutting we have is 100 hours.

01:18 So for our second equation, we can do the same thing with construction.

01:21 We have 2x again, and then y.

01:25 So 1 times y is just y and then 3 times z is equal to 100 and then same thing for the finishing we have just x plus y plus 2 z is equal to 65 so now we have a system of three equations that we can solve using elimination and our first step is going to be to subtract our second equation from our first equation so we can eliminate the 2 x variables so if we have have 2x minus 2x, the x is eliminated.

02:00 If we have 3y minus y, that gives us 2y, and then 2z minus 3z gives us minus z, and then 100 minus 100 is just 0.

02:11 And now we can take our third equation and multiply it by 2...

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