Problem 71 - College Algebra with Modeling and Visualization

Question

Technology Use technology to find the solution. A pproximate values to the nearest thousandth. $$ \begin{aligned} 0.1 x+0.3 y+1.7 z &=0.6 \\ 0.6 x+0.1 y-3.1 z &=6.2 \\ 2.4 y+0.9 z &=3.5 \end{aligned} $$

Step-by-Step Solution

Answer

\(x = -11.278\), \(y = 1.333\), \(z = 2.611\)

Step 1: Setting Up the Equation for Technology

Convert the given system of equations into a coefficient matrix for use with a calculator or software. Our system is:\[\begin{bmatrix}0.1 & 0.3 & 1.7 & | & 0.6 \0.6 & 0.1 & -3.1 & | & 6.2 \0 & 2.4 & 0.9 & | & 3.5\end{bmatrix}\]

Step 2: Inputting Data into the Calculator

Enter the matrix from Step 1 into a graphing calculator or matrix-capable app/software. Ensure values are entered accurately. Use the row augmented matrix function to input the equations into the form they appear in the matrix.

Step 3: Use Technology to Solve the Matrix

Use the calculator or software to perform Gaussian elimination or any available matrix-solving function to find the solution. The software (e.g., TI calculator, MATLAB, or Excel) will provide the values of \(x\), \(y\), and \(z\).

Step 4: Interpreting the Results

The solution to the system will be shown as a set of values for \(x\), \(y\), and \(z\). Round these values to the nearest thousandth as per the question's requirement.

Step 5: Checking the Solution

Verify the solution by substituting \(x\), \(y\), and \(z\) back into the original equations. Ensure that each equation balances to confirm the solution is correct.

Key Concepts

Matrix Solving

Matrix solving involves transforming a system of linear equations into a matrix form to make computations easier. In the matrix form, each row represents an equation, and each column corresponds to a variable. For example, in our problem, we have equations with the variables \(x\), \(y\), and \(z\). By creating a matrix that contains the coefficients of these variables, we can leverage technology or algebraic methods to solve the system efficiently.
Matrix solving becomes particularly beneficial when dealing with complex systems, as it converts the problem into an organized format that is more readily manipulated mathematically.
When using technology, it is crucial to enter all coefficients accurately into the matrix to avoid errors. This step is the foundation for using other techniques like Gaussian Elimination efficiently for finding solutions.

Gaussian Elimination

Gaussian Elimination is a systematic method used to manipulate a matrix into a row-echelon form, which simplifies the process of solving a system of equations. This technique involves a series of operations: swapping rows, multiplying a row by a constant, and adding or subtracting multiples of rows from each other.
The aim of Gaussian Elimination is to create zeros below the main diagonal of the matrix. This essentially transforms the system, making it easier to back-substitute and find the values of variables like \(x\), \(y\), and \(z\).
While the process could be manually done, using software or graphing calculators, as in the stated problem, speeds up the process significantly and minimizes errors. This makes Gaussian Elimination not only a popular method but also a highly efficient one when dealing with technical tools.

Graphing Calculator Use

Graphing calculators and other sophisticated software like MATLAB or Excel are powerful tools when it comes to solving systems of equations. For this task, you would enter the matrix formed from the coefficient matrix directly into the calculator.
Many graphing calculators have built-in functions for matrix operations. These include functions to perform Gaussian Elimination or even directly find solutions to systems. When using a calculator, it is important to follow the instructions carefully for matrix input, typically found in a manual or guide provided with the device.
Utilizing technology can greatly facilitate solving complex problems, like the given linear equations, by automating tedious calculations and providing accurate results swiftly. Just ensure each value is input precisely, and use options provided to solve and interpret results efficiently.

System of Equations

A system of equations is a set of two or more equations with the same set of variables. Solving systems involves finding the values for the variables that satisfy all of the equations simultaneously.
In this scenario, we have three equations that need to be solved for \(x\), \(y\), and \(z\). Approximating the values to the nearest thousandth ensures precision in the results, which is often necessary in practical applications where exact answers are required.
Solving a system of equations by converting it into a matrix form is a reliable method because it organizes information in a way that aligns with various solution techniques. This system-solving approach simplifies and speeds up the problem-solving process, especially when dealing with continuous variables and approximate values.