We often encounter systems of equations in which the context is not lines and planes. Economists often have to work with systems of dozens of equations involving dozens of unknowns. Relying on a geometric interpretation would not be very useful. To help organize such vast amounts of data, mathematicians have created a powerful tool called a A matrix is simply numerical data arranged in a rectangular array. We usually enclose the array in square brackets. The method of solving a linear system amounts to combining linear combinations of the equations in certain ways. Since several similar steps are involved, this method is ideally suited for technology. Solving a linear system using technology requires a systematic approach because a calculator or computer uses the same method every time. Elementary Row Operations A system of linear equations can be represented by a matrix. To obtain an equivalent system, perform any of these operations. A system of three linear equations in x, y, and 2 represents three planes in R3. It can be represented by a matrix * * * l * having the form; * * * | * , called the where each * represents a real number. * * *l * When we solve the system using matrices, we attempt to use the to obtain a l 0 0| # matrix having the form 0 1 0| # called the . The method of doing this is calledi 0 0 l | # . The matrix that results is the reduced matrix and is in Example # 1: (Solving a Matrix in R2) Solve the linear system using matrices. 6x + 10y = 14 8x6y= 20 Solution: Example # 2: (Solving 3 Matrix in R3) Solve the linear system using matrices x+3y+4z19=0 x+2y+z=12 x+y+z8=0 Solution