Problem 60 - Algebra and Trigonometry

Question

Explain how to write a linear system of three equations in three variables as a matrix equation.

Step-by-Step Solution

Answer

To write a linear system of three equations in three variables as a matrix equation, you should: arrange the equations consistently; create a 3x3 coefficient matrix where each row corresponds to an equation, and each column corresponds to a variable; create a column vector of variables, where each variable corresponds to its position across the rows of the coefficients; create a constant column vector where each constant corresponds to each equation; and finally, combine the matrices to get the final matrix equation.

Step 1: Understand the Correspondence

For a system of linear equations, each equation can be viewed as a row, and each variable can be viewed as a column. Thus, the coefficient of each variable in each equation corresponds to an entry in the matrix. Start by arranging the equations appropriately. The order of the variables should be consistent throughout all equations.

Step 2: Create The Coefficient Matrix

Next, the coefficients of the variables from all the equations should be filled into a 3x3 matrix, row by row. The order of the values should follow the order in which the variables are organized in the equations. The first row of the matrix stands for the first equation and so on.

Step 3: Formulate The Vector of Variables

The vector of variables can be formed by writing all the variables in the order in which they appear in the equations. This creates a column matrix, also known as a column vector.

Step 4: Create the Constant Matrix

The constant matrix or the column vector is created by writing down the constants on the right side of each equation. Each constant in the order corresponds to each equation.

Step 5: Combine into a Matrix Equation

Finally, formulate the matrix equation by combining the coefficient matrix, vector of variables, and constant matrix. The relation of a matrix equation is: [Coefficient Matrix] x [Variable Vector] = [Constant Matrix].