Problem 52 - College Algebra

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 system of linear equations as a matrix equation, place the coefficients of the equations in a matrix \(A\), the variables in a vector \(X\), and the constants in a vector \(B\). The resulting matrix equation is \(AX = B\).

Step 1: Definition of Linear system of equations

A linear system of three equations in three variables is a set of equations of the form: \(ax + by + cz = d\) \(ex + fy + gz = h\) \(ix + jy + kz = l\) where \(x, y, z\) are the variables, and \(a, b, c, d, e, f, g, h, i, j, k, l\) are coefficients.

Step 2: Matrix representation

The three equations can be rewritten as a single matrix equation, \(AX = B\), where \(A\) = \(\begin{bmatrix} a & b & c \ e & f & g \ i & j & k \end{bmatrix}\), \(X\) = \(\begin{bmatrix} x \ y \ z \end{bmatrix}\) and \(B\) = \(\begin{bmatrix} d \ h \ l \end{bmatrix}\).

Step 3: Formulation of Matrix Equation

Thus, the matrix equation representing the system of equations would be, \(\begin{bmatrix} a & b & c \ e & f & g \ i & j & k \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} d \ h \ l \end{bmatrix}\).