Problem 25 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the system of equations using (a) Gaussian elimination and (b) Cramer's Rule. Which method do you prefer, and why? \(\left\\{\begin{array}{l}3 x+3 y+5 z=1 \\ 3 x+5 y+9 z=2 \\ 5 x+9 y+17 z=4\end{array}\right.\)

Step-by-Step Solution

Answer

The solutions obtained by Gaussian Elimination and Cramer's Rule should be the same. The preference between these two methods greatly depends on the particular system of equations presented and the individual's comfort level with each method. Gaussian elimination can be easier to understand and apply, especially for larger systems, but it also may involve more computational steps. On the other hand, Cramer's Rule can be faster for smaller systems but involves the calculation of determinants, which can be cumbersome.

Step 1: Solving Using Gaussian Elimination

Gaussian elimination involves transforming the system of equations into a simple upper triangular form through row operations. Then, use back substitution to solve for the unknowns. 1. Write the system of linear equations in augmented matrix form: \[\left[\begin{array}{ccc|c}3 & 3 & 5 & 1 \3 & 5 & 9 & 2 \5 & 9 & 17 & 4\end{array}\right]\]2. Use row operations to achieve upper triangular form. For this system, subtract row 1 from row 2 and subtract twice row 1 from row 3:\[\left[\begin{array}{ccc|c}3 & 3 & 5 & 1 \0 & 2 & 4 & 1 \-1 & 3 & 7 & 2\end{array}\right]\]3. Finally, we can use back substitution to solve for the unknowns \(x\), \(y\), and \(z\). You may need to perform some additional row operations to get zeros in all the right places.

Step 2: Solving Using Cramer's Rule

Cramer's Rule makes use of determinants to solve the system of equations. 1. First, compute the determinant of the coefficient matrix (\(D\)).2. Then find the determinants of matrices obtained by replacing each column in turn with the column vector of solution values (\(D_x, D_y, D_z\)).3. The solutions can then be given by \(x=D_x/D\), \(y=D_y/D\), and \(z=D_z/D\).4. Calculate these determinants and find the solutions.

Step 3: Comparison of the Methods

The comparison should be made based on how easy or difficult it was to perform each method. Consider aspects such as the number of computations required, the ease of performing the calculations and whether or not one would be less prone to making errors with one method over another.

Key Concepts

Gaussian Elimination

Gaussian elimination is a method used to solve systems of linear equations. It transforms the equations to an upper triangular form, which makes them easier to solve using back substitution. This process involves performing row operations on the matrix formed by the coefficients of the variables and the constants on the right side of the equations.

• First, we represent the system in augmented matrix form, which includes the coefficient matrix and the constant terms.
• The goal is to manipulate this matrix so that the entries below the main diagonal are all zeros. This is done through a series of steps: swapping rows, multiplying a row by a nonzero scalar, and adding or subtracting rows.
• Once the matrix is in an upper triangular form, back substitution can be used to find the values of the variables.

Gaussian elimination is straightforward for most systems. It allows us to solve the equations progressively, starting from the bottom row and moving upwards. This method works best with smaller systems, as large matrices may lead to extensive computations.

Cramer's Rule

Cramer's Rule is another technique for solving systems of linear equations, especially when the number of equations matches the number of unknowns. This approach uses determinants, which are special numbers calculated from a square matrix.

• To use Cramer's Rule, the coefficient matrix of the system must have a non-zero determinant.
• First, compute the determinant of the original coefficient matrix, labeled as \(D\).
• For each variable, replace the corresponding column of the coefficient matrix with the constants on the right side of the equations. Compute the determinant of each new matrix, giving \(D_x\), \(D_y\), and \(D_z\).
• The solutions are then given by dividing each of these determinants by \(D\): \(x = D_x / D\), \(y = D_y / D\), and \(z = D_z / D\).

Cramer's Rule is computationally simple and provides direct solutions. However, for large systems or when the determinant of the matrix is zero, it cannot be used.

Augmented Matrix

An augmented matrix is a compact way to represent a system of linear equations. It combines the coefficient matrix and the constants into a single matrix form, separated by a vertical line.

• The left side of the augmented matrix includes all coefficients of the variables from the system of equations.
• The right side consists of the constants from the equations.
• For example, the system of equations \(\begin{align*} 3x + 3y + 5z & = 1 \ 3x + 5y + 9z & = 2 \ 5x + 9y + 17z & = 4 \\end{align*}\)is represented by the augmented matrix:\[\begin{bmatrix} 3 & 3 & 5 & \vert & 1 \3 & 5 & 9 & \vert & 2 \5 & 9 & 17 & \vert & 4 \end{bmatrix}\]

Using augmented matrices streamlines operations in methods like Gaussian elimination. It is essential to keep track of the constant terms when altering the coefficients during row operations.

Back Substitution

Back substitution is the final phase of solving a system of equations that has been transformed into an upper triangular form through Gaussian elimination.

• After achieving an upper triangular matrix, the system can be solved easily starting from the last row. That row typically involves only the last variable, making it straightforward to solve.
• Once the last variable is determined, substitute this value into the row above it to find the second-last variable.
• Continue this process upwards until all variables have been found.

This method ensures that each variable is solved one step at a time, leveraging the simplicity of an upper triangular system. Back substitution simplifies the solving process, making it a reliable and efficient technique for systems that have undergone Gaussian elimination.