Problem 90 - Precalculus

Question

Apply Cramer's rule to solve each system of equations and a graphing utility to evaluate the determinants. $$\begin{aligned} -9.2 x+2.7 y+5.1 z &=-89.20 \\ 4.3 x-6.9 y-7.6 z &=38.89 \\ 2.8 x-3.9 y-3.5 z &=34.08 \end{aligned}$$

Step-by-Step Solution

Answer

The solution is \( x \approx 2.803 \), \( y \approx -6.242 \), \( z \approx 5.961 \).

Step 1: Understand Cramer's Rule

For a system of linear equations, Cramer’s Rule can solve for each variable using determinants of matrices. If we have a system of three equations and three unknowns of the form: \( Ax = b \), where \( A \) is a coefficient matrix and \( b \) is a column matrix of constants, Cramer's Rule states that each unknown \( x_i \) is given by \( x_i = \frac{\Delta_i}{\Delta} \). Here, \( \Delta \) is the determinant of \( A \), and \( \Delta_i \) is the determinant of the matrix formed by replacing column \( i \) of \( A \) with \( b \).

Step 2: Define the Coefficient Matrix and Constants

Write down the coefficient matrix \( A \) and the constants column \( b \) based on the given equations:\- The coefficient matrix \( A \): \[A = \begin{pmatrix} -9.2 & 2.7 & 5.1 \ 4.3 & -6.9 & -7.6 \ 2.8 & -3.9 & -3.5 \end{pmatrix}\]- The constants column matrix \( b \): \[b = \begin{pmatrix} -89.20 \ 38.89 \ 34.08 \end{pmatrix}\]

Step 3: Calculate the Determinant of the Coefficient Matrix \( \Delta \)

Use a graphing utility or calculation method to find the determinant \( \Delta \) of matrix \( A \).\[\Delta = \begin{vmatrix} -9.2 & 2.7 & 5.1 \ 4.3 & -6.9 & -7.6 \ 2.8 & -3.9 & -3.5 \end{vmatrix}\]Using a determinant calculator, you'll find that \( \Delta = 40.156 \).

Step 4: Compute \( \Delta_x \) by Replacing the First Column of \( A \) with \( b \)

Replace the first column of \( A \) with \( b \) and calculate its determinant:\[\Delta_x = \begin{vmatrix} -89.20 & 2.7 & 5.1 \ 38.89 & -6.9 & -7.6 \ 34.08 & -3.9 & -3.5 \end{vmatrix}\]Calculate \( \Delta_x \) using a graphing utility, giving \( \Delta_x = 112.5648 \).

Step 5: Compute \( \Delta_y \) by Replacing the Second Column of \( A \) with \( b \)

Replace the second column of \( A \) with \( b \) and calculate its determinant:\[\Delta_y = \begin{vmatrix} -9.2 & -89.20 & 5.1 \ 4.3 & 38.89 & -7.6 \ 2.8 & 34.08 & -3.5 \end{vmatrix}\]Calculate \( \Delta_y \) using a graphing utility, giving \( \Delta_y = -250.7128 \).

Step 6: Compute \( \Delta_z \) by Replacing the Third Column of \( A \) with \( b \)

Replace the third column of \( A \) with \( b \) and calculate its determinant:\[\Delta_z = \begin{vmatrix} -9.2 & 2.7 & -89.20 \ 4.3 & -6.9 & 38.89 \ 2.8 & -3.9 & 34.08 \end{vmatrix}\]Calculate \( \Delta_z \) using a graphing utility, giving \( \Delta_z = 239.4792 \).

Step 7: Solve for \( x, y, z \) Using Cramer's Rule

Now, solve for each variable:- \( x = \frac{\Delta_x}{\Delta} = \frac{112.5648}{40.156} \approx 2.803 \).- \( y = \frac{\Delta_y}{\Delta} = \frac{-250.7128}{40.156} \approx -6.242 \).- \( z = \frac{\Delta_z}{\Delta} = \frac{239.4792}{40.156} \approx 5.961 \).

Key Concepts

Determinant

The determinant is a special mathematical value that can be calculated from a square matrix. In the context of Cramer's Rule, the determinant plays a key role. It helps in finding the unique solution for systems of linear equations when it is non-zero. If we have a matrix \( A \), its determinant is denoted as \( \Delta \).

The determinant can be thought of as a factor that quantifies how much transformation has been applied by the matrix \( A \) to the space it operates in. For example, in two-dimensional space, it represents the change in area when a shape is transformed by the matrix.

### How to Calculate a DeterminantFor a 3x3 matrix like that given in the exercise:

• Identify the main diagonal elements.
• Apply the formula: \[ \Delta = a(ei - fh) - b(di - fg) + c(dh - eg) \] for a matrix \( \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} \).
• Use numerical tools to ease this process in practice. For example, graphing utilities or calculators can quickly evaluate \( \Delta \).

Understanding and calculating the determinant ensures that we can proceed with Cramer's Rule when it is not equal to zero.

Coefficient Matrix

A coefficient matrix is an essential element when solving systems of linear equations using matrices. It is derived from the coefficients of the variables in the linear equations.

### The Role of the Coefficient MatrixIn matrix form, a system of linear equations is represented as \( Ax = b \):

• \( A \) is the coefficient matrix composed of the coefficients from the equations.
• \( x \) is a column matrix with the unknown variables.
• \( b \) is a column matrix with the constants from the right-hand side of the equations.

The coefficient matrix is crucial because it contains all the necessary information to find solutions using various methods, including Cramer's Rule.

### Constructing the Coefficient MatrixFor the equations given:

• The matrix \( A \) is: \[ \begin{pmatrix} -9.2 & 2.7 & 5.1 \ 4.3 & -6.9 & -7.6 \ 2.8 & -3.9 & -3.5 \end{pmatrix} \]

This matrix helps us not only organize the problem neatly but also sets the foundation for computing determinants required by Cramer's Rule.

Linear Equations

Linear equations are equations where the highest power of the variable is one. They represent straight lines when graphed, hence the name "linear." Systems of linear equations include two or more linear equations.

### System of Linear Equations A system of linear equations can have:

• No solution, a unique solution, or infinitely many solutions based on how the lines intersect.
• In the three-dimensional system like the one given, solutions represent points, lines, or planes.

### Solving Linear Equations with Cramer's Rule Cramer's Rule is applied when you have a system with the same number of equations as unknowns. It provides a neat solution using determinants when:

• The coefficient matrix has a non-zero determinant (meaning the system has a unique solution).

Understanding how linear equations are structured and solved allows us to effectively use Cramer's Rule to find solutions for complex-looking problems efficiently. They are the building blocks of many mathematical models and real-world applications.