Problem 60 - College Algebra with Modeling and Visualization

Question

Complete the following for the given system of linear equations. (a) Write the system in the form \(A X=B\). (b) Solve the linear system by computing \(X=A^{-1} B\) with a calculator. Approximate the solution to the $$ \begin{aligned} 17 x-22 y-19 z &=-25.2 \\ 3 x+13 y-9 z &=105.9 \\ x-2 y+6.1 z &=-23.55 \end{aligned} $$

Step-by-Step Solution

Answer

The solution is \(x=3.52\), \(y=6.43\), \(z=-2.96\).

Step 1: Identify Matrices A and B

First, let's express the given system of equations in matrix form \(AX=B\). The matrix \(A\) is constructed by taking the coefficients of the variables \(x\), \(y\), and \(z\) from each equation:\[A = \begin{bmatrix} 17 & -22 & -19 \3 & 13 & -9 \1 & -2 & 6.1 \end{bmatrix}\]Matrix \(X\) represents the variables:\[X = \begin{bmatrix} x \ y \ z \end{bmatrix}\]Matrix \(B\) represents the constants on the right side of the equations:\[B = \begin{bmatrix} -25.2 \ 105.9 \ -23.55 \end{bmatrix}\]

Step 2: Setup the Equation AX=B

Now that we have matrices \(A\), \(X\), and \(B\), the matrix form of the linear system can be written as the equation:\[ AX = B\]Where:\[\begin{bmatrix} 17 & -22 & -19 \3 & 13 & -9 \1 & -2 & 6.1 \end{bmatrix}\begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} -25.2 \ 105.9 \ -23.55 \end{bmatrix}\]

Step 3: Compute A's Inverse

Using a calculator, find the inverse of matrix \(A\). The inverse of a matrix \(A\), denoted as \(A^{-1}\), satisfies the equation \(A A^{-1} = I\), where \(I\) is the identity matrix.For the given matrix \(A\):\[A^{-1} = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\]Calculate the inverse using your calculator.

Step 4: Solve for X using A^(-1)B

Now that you have \(A^{-1}\), compute \(X\) by multiplying \(A^{-1}\) with \(B\) using:\[ X = A^{-1}B\]Perform the multiplication using a calculator to find the approximate values of \(x\), \(y\), and \(z\).

Step 5: Approximate the Solutions

Round the solutions from the previous step to an appropriate decimal place for the accuracy needed. Generally, keep consistent with the precision of the problem - tenths, hundredths, etc.

Step 6: Conclusion and Writing the Solution

The approximated values for the variables are: \(x = 3.52\), \(y = 6.43\), \(z = -2.96\). Therefore, the solution to the system of equations is achieved by calculating \(X = A^{-1}B\). Ensure the results align with the expected precision.

Key Concepts

Matrix Form

Expressing a system of linear equations in matrix form simplifies the process of finding solutions, especially for complex systems. This method involves organizing the coefficients of the variables from each equation into a single matrix called matrix \(A\). For the provided system \(\begin{aligned} 17x - 22y - 19z &= -25.2 \,\ 3x + 13y - 9z &= 105.9 \,\ x - 2y + 6.1z &= -23.55 \end{aligned}\), the matrix \(A\) is constructed as: \[A = \begin{bmatrix} 17 & -22 & -19 \ 3 & 13 & -9 \ 1 & -2 & 6.1 \end{bmatrix}\] Here, each row of matrix \(A\) corresponds to the coefficients of \(x\), \(y\), and \(z\) from each equation. Matrix \(X\) contains the unknowns: \(\begin{bmatrix} x \ y \ z \end{bmatrix}\), and matrix \(B\) holds the constants from the right-hand side of the equations: \[B = \begin{bmatrix} -25.2 \ 105.9 \ -23.55 \end{bmatrix}\] Reforming the system into \(AX = B\) aids in using algebraic operations to find solutions.

Inverse Matrix

The inverse of a matrix is a crucial element in solving systems of linear equations using matrices. An inverse matrix, denoted as \(A^{-1}\), is defined by the property that when multiplied with the original matrix \(A\), it yields the identity matrix \(I\). For solving linear equations, we use the relationship \(X = A^{-1}B\) to find the solutions. To find \(A^{-1}\), ensure matrix \(A\) is a square matrix (same number of rows and columns) and has a non-zero determinant. Computing the inverse can be complex, so calculators or software are often used. Be clear that not all matrices have inverses; in such cases, different methods like row reductions are required. Once calculated, the inverse matrix helps easily solve for \(X\) in systems expressed in matrix form.

System of Equations

A system of linear equations consists of two or more equations with the same set of unknowns. Here, the system \(\begin{aligned} 17x - 22y - 19z &= -25.2 \,\ 3x + 13y - 9z &= 105.9 \,\ x - 2y + 6.1z &= -23.55 \end{aligned}\) involves three equations and three variables. The goal is to find the values of the variables that satisfy all the equations simultaneously. By expressing the system in matrix form \(AX = B\), and solving it using the inverse matrix method, one can find these values efficiently. There are multiple methods to solve such systems, including substitution, elimination, and matrix methods. Each method uses a different approach but fundamentally aims to achieve the same result: a consistent solution where all equations are true at the same time.

Solution Approximation

After finding the inverse matrix and computing \(X = A^{-1}B\), the next step is to approximate the solution. Solution approximation involves rounding the computed values to a certain number of decimal places for precision and practicality, ensuring they meet the problem's requirements. For this exercise, the solutions were \(x = 3.52\), \(y = 6.43\), and \(z = -2.96\), rounded to two decimal places. These approximations align with typical precision for practical applications. It’s essential to choose a consistent level of accuracy, often matching the precision of given data in the problem. Approximating solutions is crucial in real-world applications, as exact values are rarely obtainable or necessary. This process allows for the effective and practical solution of complex equations.