Solving a System of Three Equations in Three Variables by Elimination

There will always be several choices as to where to begin, but the most obvious first step here is to eliminate $x$ by adding equations (1) and (2).

The second step is multiplying equation (1) by $\mathrm{-2}$ and adding the result to equation (3). These two steps will eliminate the variable $x.$

In equations (4) and (5), we have created a new two-by-two system. We can solve for $z$ by adding the two equations.

Choosing one equation from each new system, we obtain the upper triangular form:

Next, we back-substitute $z=2$ into equation (4) and solve for $y.$

Finally, we can back-substitute $z=2$ and $y=\mathrm{-1}$ into equation (1). This will yield the solution for $x.$

The solution is the ordered triple $\left(1,\mathrm{-1},2\right).$ See Figure 4.