Most interesting problems (refer to the Sample Word Problems Spreadsheet) involve finding multiple values that satisfy multiple linear equations. Not everything in life is linear, but often it is at least APPROXIMATELY linear close to the point we are interested in. The equations are known relationships among the various quantities being sought (these quantities are typically referred to as "variables", while the actual value you are looking for is called the "solution"). Typically you need to have as many equations as there are variables, however, even if you have fewer equations you can at least narrow your answer down to a certain "family" of solutions.

Two linear equations are demonstrated in the Locating an Intersection Spreadsheet as two lines. If the lines are parallel and disjoint, then they are said to be "inconsistent" and there is no intersection, thus no values of x and y satisfy both equations and there is no solution. If the lines are parallel and have the same y-intercept then they are the same line and every point on the line is a solution. If the lines are not parallel, then they will only intersect in one place and there will only be one solution.

Three linear equations are demonstrated in this Three Intersecting Planes VRML document (since it is in 3D you need a VRML plugin to view it). You can generate your own three intersecting planes VRML document using the VRML Generation Spreadsheet. Similar to two intersecting lines, there is typically only one point in space where three planes come together (unless two of them are parallel - no intersection, or one plane is a combination of the other two - they come together along a whole 3D line).

What we are concerned with here is not how you come up with the equations (Word Problem Hints), but how you solve them. Technically the process isn't any different than solving a single equation, you just end up doing it repeatedly. If you solve for one variable in terms of the other in one of the equations you can then substitute this expression in the remaining equations, giving you one fewer equations and one fewer variables. You repeat this process with the remaining equations until you are down to only one variable in one equation, for which you can obtain a solution. Then you work your way backwards by substituting each solution you find into the previous equations so that you obtain additional solutions.

In general this takes quite a bit of elbow grease. To keep track of the calculations in an organized way the process of "left-to-right elimination" has been developed. This has been further streamlined by reducing the equations to matrices of coefficients (numbers multiplying the variables) and constants (numbers without variables on the opposite side of the "=" sign) and doing Gauss-Jordan elimination. Most graphing calculators have matrix operations, including "rref" (reduced row echelon form) that performs Gauss-Jordan elimination automatically. If your calculator does not, you can use this Gauss-Jordan Elimination Spreadsheet to solve systems of three equations and three variables. A more general purpose and somewhat easier to use "rref" calculator is this java applet.

In the case where there is a unique solution, Gauss-Jordan elimination is basically the process of reducing the coefficient matrix to an identity matrix by modifying the constant matrix. This is equivalent to dividing out the coefficient in a single equation with one variable and can be represented as multiplying the system by the inverse matrix of the coefficient matrix. This matrix multiplication is relatively simple, but finding the inverse manually would still require Gauss-Jordan elimination. Fortunately this process is automated on most graphing calculators and Excel. It is demonstrated in this Solution Using an Inverse Spreadsheet.

Contact Leo Wibberly at ldwibber@vcu.edu or (804) 740-4650 to make appointments