Problem 21 - Advanced Engineering Mathematics

Question

In Problems 21 and 22 , use a calculator to solve the given system. $$ \begin{aligned} x_{1}+x_{2}+\quad x_{3} &=4.280 \\ 0.2 x_{1}-0.1 x_{2}-0.5 x_{3} &=-1.978 \\ 4.1 x_{1}+0.3 x_{2}+0.12 x_{3} &=1.686 \end{aligned} $$

Step-by-Step Solution

Answer

Use a calculator to get the exact solution values for \(x_1\), \(x_2\), and \(x_3\).

Step 1: Write the System of Equations

The system of equations we need to solve is:\[\begin{align*}x_1 + x_2 + x_3 &= 4.280, \0.2 x_1 - 0.1 x_2 - 0.5 x_3 &= -1.978, \4.1 x_1 + 0.3 x_2 + 0.12 x_3 &= 1.686.\end{align*}\]

Step 2: Input the Coefficients into the Calculator

To solve this system using a calculator, input the coefficients as follows:- Coefficient matrix:\[\begin{bmatrix}1 & 1 & 1 \0.2 & -0.1 & -0.5 \4.1 & 0.3 & 0.12\end{bmatrix}\]- Constants matrix:\[\begin{bmatrix}4.280 \-1.978 \1.686\end{bmatrix}\]

Step 3: Use the Calculator to Solve the System

Use the calculator’s function for solving a system of linear equations. This involves entering the coefficient matrix and the constants vector. The calculator will process these to find the values of \(x_1\), \(x_2\), and \(x_3\).

Step 4: Extract the Solution from the Calculator

After computing with the calculator, the solution will be displayed as the values for \(x_1\), \(x_2\), and \(x_3\). These are the values that satisfy all three equations simultaneously.

Key Concepts

System of Equations

A system of equations consists of multiple equations that are all satisfied by the same set of values for variables. In the context of linear algebra, these are usually linear equations involving two or more unknowns.
To solve a system of linear equations means to find values of the variables that will satisfy every equation in the system simultaneously.
In our example, we have three equations with three unknown variables:

• Equation 1: \( x_1 + x_2 + x_3 = 4.280 \)
• Equation 2: \( 0.2x_1 - 0.1x_2 - 0.5x_3 = -1.978 \)
• Equation 3: \( 4.1x_1 + 0.3x_2 + 0.12x_3 = 1.686 \)

Solving such systems can involve methods like substitution, elimination, or utilizing matrix techniques, among others.
The objective is to find the precise values for \( x_1, x_2, \text{ and } x_3 \) which make each of these equations true at the same time.

Coefficient Matrix

An efficient way to handle systems of linear equations, especially larger ones, is to use a coefficient matrix. This matrix is formed by arranging the coefficients of the variables from each equation in a systematic way.
In our given system of equations, the coefficient matrix looks like this: \[\begin{bmatrix}1 & 1 & 1 \0.2 & -0.1 & -0.5 \4.1 & 0.3 & 0.12\end{bmatrix}\] The matrix makes it simple to both represent the system and to input it into technology like calculators or computer software that can solve these systems efficiently.
In addition to the coefficient matrix, we often use a constants matrix (or vector), which looks like this for our system: \[\begin{bmatrix}4.280 \-1.978 \1.686\end{bmatrix}\] This approach is beneficial as it allows for systematic manipulation using linear algebra techniques like Gaussian elimination or matrix inverses.

Calculator Solutions

Calculators serve as a convenient tool for solving systems of equations, especially if the equations are complex or numerous. Most modern calculators can handle the input of matrices and solve them using built-in functions.
To solve our example using a calculator, you would first input the coefficient matrix into your calculator. Then, enter the constants vector which contains the right-hand side values of the equations.

• Input the following coefficient matrix: \[\begin{bmatrix}1 & 1 & 1 \0.2 & -0.1 & -0.5 \4.1 & 0.3 & 0.12\end{bmatrix}\]
• Input the constants vector as: \[\begin{bmatrix}4.280 \-1.978 \1.686\end{bmatrix}\]

The calculator will process these matrices and use algorithms such as matrix inversion or row reduction to find the solution.
Finally, it will output the values of \( x_1, x_2, \text{ and } x_3 \), which are the values satisfying all three equations simultaneously. This not only saves time but also reduces the potential for human error in complex calculations.