Problem 69 - College Algebra with Modeling and Visualization

Question

Technology Use technology to find the solution. A pproximate values to the nearest thousandth. $$ \begin{array}{l} 2.1 x+0.5 y+1.7 z=4.9 \\ -2 x+1.5 y-1.7 z=3.1 \\ 5.8 x-4.6 y+0.8 z=9.3 \end{array} $$

Step-by-Step Solution

Answer

The approximate solution is \(x = 2.337\), \(y = 1.238\), \(z = 1.431\).

Step 1: Input System into Technology

The given system of linear equations is: \(2.1x+0.5y+1.7z=4.9\), \(-2x+1.5y-1.7z=3.1\), and \(5.8x-4.6y+0.8z=9.3\). Input these equations into a computational tool or software, such as a graphing calculator or a computer algebra system capable of solving systems of equations.

Step 2: Use Tool to Solve System

Utilize the tool's function or algorithm to solve the system of equations. Most software provides a built-in function for solving systems of linear equations, which determines the values of \(x\), \(y\), and \(z\) that satisfy all three equations simultaneously.

Step 3: Extract and Approximate Solution

After computing, extract the solution values for \(x\), \(y\), and \(z\) from the software's output. Round each value to the nearest thousandth as required. For example, if \(x = 1.123456\), round it to \(x = 1.123\).

Step 4: Verify Solution

To ensure accuracy, substitute the approximate solution values back into the original equations to confirm that they are satisfied within rounding error.

Key Concepts

Understanding Systems of Equations

A system of equations is a set of two or more equations that have multiple variables. These equations are solved together because they share the same variables, and the solution to the system is the set of values that satisfies all equations simultaneously. When dealing with a system of linear equations, like the ones in our exercise, each equation represents a plane in three-dimensional space. The point of intersection of these planes is the solution of the system if it exists.

• Each variable in the system typically corresponds to a dimension in geometry.
• The solution can be a point, a line, or in some cases, there might be no solution or infinite solutions if the planes are parallel or coincide.

Understanding how systems of equations interact helps us model real-world problems, making them a powerful tool in science and engineering.

Role of Computation Tools

Computation tools are technological resources that can be used to tackle mathematical problems. These include software like graphing calculators, computer algebra systems, and various mathematical software applications. In the context of solving systems of linear equations, these tools automate the complex algebraic manipulations needed to solve for the unknown variables.

• Computation tools can efficiently handle large and complex systems.
• Their algorithms are designed to find accurate solutions quickly.

Using these tools allows students to focus on understanding and interpreting the results rather than getting bogged down in computations.

Basic Algebra Concepts in Systems

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols. In systems of equations, algebra is crucial as it provides the method for describing mathematical relationships. Solving systems of equations typically involves algebraic techniques such as substitution, elimination, or matrix operations.

• Substitution involves solving one equation for one variable and then substituting that expression in the other equations.
• Elimination involves adding or subtracting equations to eliminate one variable, simplifying the system to make it easier to solve.

Understanding these algebraic methods allows students to bring theoretical solutions into practical applications, whether manually or using technology.

Graphing Calculator as a Solving Aid

A graphing calculator is a handheld computational device capable of plotting graphs, solving equations, and performing various types of mathematical calculations. In the context of systems of equations, graphing calculators can be particularly useful for visualizing the relationships between the equations and for finding exact or approximate solutions.

• Graphing calculators can plot each equation as a graph and show their point of intersection, which is the solution.
• They contain built-in functions for solving systems of equations, making them a convenient tool for students.

Employing a graphing calculator not only aids in solving but also enhances comprehension by allowing students to visualize how equations and their solutions behave in space.