Technical Mathematics, Sixth Edition

9.4. Systems of Three Equations

Our strategy here is to reduce a given system of three equations in three unknowns to a system of two equations in two unknowns, which we already know how to solve.

9.4.1. Addition-Subtraction Method

We take any two of the given equations and, by addition-subtraction or substitution, eliminate one variable, obtaining a single equation in two unknowns. We then take another pair of equations (which must include the one not yet used, as well as one of those already used) and similarly obtain a second equation in the same two unknowns. This pair of equations can then be solved simultaneously, and the values obtained substituted back to obtain the third variable.

Example 26: Solve the following: Solution: It is a good idea to number your equations, as in this example, to help keep track of your work. Let us start by eliminating x from Eqs. 1 and 2. We now eliminate the same variable, x, from Eqs. 1 and 3. Now we solve Eqs. 6 and 9 simultaneously. Substituting z = −1 into Eq. 6 gives us Substituting y = 2 and z = −1 into Eq. 1 yields Our solution is ...