Problem 59 - Algebra and Trigonometry

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

Apply Cramer's Rule to the system of equations by first setting up the coefficient matrix and finding its determinant. Then, substitute the columns of the coefficient matrix with the constants of the equations, and find the determinant each time. Divide these determinants by the determinant of the coefficient matrix to find the values of the variables.

Step 1: Set Up the Coefficient Matrix and Compute its Determinant

Create a 3x3 matrix using the coefficients of the variables (\(a, b, c, e, f, g, i, j, k\)). The determinant of this matrix is denoted as \(D\). Use any method of your choice to find the determinant, e.g. minors and cofactors method.

Step 2: Compute the Determinants for x, y, z

Replace the first, second, and third columns of the original matrix with the constants on the right side of the equations (d, h, l) one at a time, and compute the determinants \(D_x\), \(D_y\), and \(D_z\) respectively.

Step 3: Apply Cramer's Rule

Finally, apply Cramer's Rule by dividing the determinants \(D_x, D_y, D_z\) by the determinant \(D\). This results in solutions for the variables \(x, y, z\): \(x = D_x/D, y = D_y/D, z = D_z/D\).