Question

Find the solution of the given problem by: (a) creating an appropriate system of linear equations (b) forming the augmented matrix that corresponds to this system (c) putting the augmented matrix into reduced row echelon form (d) interpreting the reduced row echelon form of the matrix as a solution. A carpenter can make two sizes of table, grande and venti. The grande table requires 4 table legs and 1 table top; the venti requires 6 table legs and 2 table tops. After doing work, he counts up spare parts in his warehouse and realizes that he has 86 table tops left over, and 300 legs. How many tables of each kind can he build and use up exactly all of his materials?

Short answer

42 grande tables and 22 venti tables.

Step-by-step solution

Define Variables

Let's define the variables for the problem: Let \( x \) be the number of grande tables and \( y \) be the number of venti tables that the carpenter can make.

Set Up System of Equations

Using the problem statement, we can write two equations to express the constraints on table legs and table tops.For table tops: \[ x + 2y = 86 \] For table legs: \[ 4x + 6y = 300 \]

Form the Augmented Matrix

We create an augmented matrix from the system of equations:\[\begin{bmatrix}1 & 2 & | & 86 \4 & 6 & | & 300 \\end{bmatrix}\]

Apply Row Operations to Achieve Row Echelon Form

First, eliminate the first element of the second row by performing the row operation: \( R_2 = R_2 - 4R_1 \).The resulting matrix is:\[\begin{bmatrix}1 & 2 & | & 86 \0 & -2 & | & -44 \\end{bmatrix}\]

Achieve Reduced Row Echelon Form

To turn the matrix into reduced row-echelon form, divide the second row by \(-2\): \( R_2 = -\frac{1}{2}R_2 \).This results in: \[\begin{bmatrix}1 & 2 & | & 86 \0 & 1 & | & 22 \\end{bmatrix}\]Finally, apply \( R_1 = R_1 - 2R_2 \) to eliminate the second element of the first row:\[\begin{bmatrix}1 & 0 & | & 42 \0 & 1 & | & 22 \\end{bmatrix}\]

Interpret the Solution

The matrix now represents the equations:\[ x = 42 \] \[ y = 22 \]This implies the carpenter can make 42 grande tables and 22 venti tables using exactly all available materials.

Key concepts

Systems of Linear Equations

A system of linear equations consists of multiple linear equations that share the same set of variables. These are equations that we attempt to solve simultaneously because they are interrelated via the variables. In this context, linear refers to straight-line relationships defined by these equations.

• Each equation in a system describes a linear relationship between variables, typically depicted as a line in a two-dimensional space.
• The goal is to find the values of the variables that satisfy all equations in the system simultaneously.

For our carpenter's problem, the system of linear equations is constructed based on the inventory constraints. Specifically, one equation (\( x + 2y = 86 \)) utilizes the available table tops, and another (\( 4x + 6y = 300 \)) uses the available table legs.
The variables, in this case, represent the quantities of different types of tables the carpenter can build.

Augmented Matrices

An augmented matrix is a practical tool for solving systems of linear equations. It merges the coefficients of variables and the constant terms into one streamlined, matrix format. This representation is especially useful for applying matrix operations.

• The left-hand side of the augmented matrix consists of the coefficients of the variables in the system.
• The right-hand side holds the constants from each equation after the equal sign, separated by a vertical bar.

In our example, the system \( x + 2y = 86 \) and \( 4x + 6y = 300 \) transforms into the matrix:
\[\begin{bmatrix}1 & 2 & | & 86 \4 & 6 & | & 300 \\end{bmatrix}\]
This matrix allows us to perform operations that simplify the system into forms where solutions become obvious, using concepts such as the reduced row echelon form.

Reduced Row Echelon Form

The reduced row echelon form (RREF) of a matrix is a simplified version that directly shows the solutions to a system of equations if they exist. This form is achieved through a series of row operations and follows specific criteria:

• The first non-zero element in each row is 1, often called the leading 1.
• Each leading 1 is the only non-zero entry in its column.
• Each leading 1 appears to the right of any leading 1s in earlier rows.

To convert our augmented matrix to RREF, we perform row operations. First, eliminate any leading coefficients, and then scale pivot rows: \[\begin{bmatrix}1 & 0 & | & 42 \0 & 1 & | & 22 \\end{bmatrix}\]
This matrix equates to the system: \( x = 42 \) and \( y = 22 \). Thus, the carpenter can precisely construct 42 grande tables and 22 venti tables by using all his materials.