Problem 59 - College Algebra

Question

Without going into too much detail, describe how to solve a linear system in three variables using Cramer's Rule.

Step-by-Step Solution

Answer

To use Cramer's rule in solving a linear system with three variables, write down the system and form a coefficient matrix. Calculate the determinant of this matrix. Create minor matrices by replacing one column at a time with the constants from the equations. Calculate the determinant for each minor matrix, then solve for each variable by dividing the determinant of the respective minor matrix by the determinant of the coefficient matrix.

Step 1: Write down the system

To start with, write down the system of equations that is to be solved. Supposing the system has the form: \(ax + by + cz = j\), \(dx + ey + fz = k\), and \(gx + hy + iz = l\)

Step 2: Write down the coefficient matrix

The coefficient matrix \(A\) is formed by the coefficients of the variables in the system of equations. In this case it is \[A= \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\]

Step 3: Calculate the determinant of A

The determinant \(D\) is calculated by the coefficients of the variables from the system of equations. Use the determinant formula for a 3x3 matrix.

Step 4: Minor Matrices, or Variable Matrices

We'll need to create new matrices by substituting our answers \(j, k, l\), one column at a time.

Step 5: Calculate the determinants

For each of the matrices from step 4, calculate the determinant. Be careful to substitute in the correct column depending on the variable being solved for

Step 6: Solve for the variables using Cramer's Rule

Using Cramer's Rule, each of the variable \(x, y, z\) is given as the ratio of the determinant of the respective minor matrix to the determinant of the coefficient matrix \(D\). For example, \( x = \frac{D_x}{D}, y = \frac{D_y}{D}, z = \frac{D_z}{D} \)