How Do You Solve a System of Three Linear Equations in Three Unknowns?

Solve the system by using elimination to reduce three equations in three unknowns to two equations in two unknowns. Then solve the reduced system, back-substitute, and check the solution in the original equations.

This is IB Mathematics: Analysis and Approaches HL, sub-topic AHL 1.16: Solution of systems of linear equations. The algebraic method below is suitable for Paper 1; matrix or row-reduction technology may also be used on Paper 2.

Elimination works because replacing one equation with a linear combination of existing equations does not change the common solution set. Choosing the same variable for both eliminations creates a smaller system efficiently and reduces sign errors.

Elimination Method

Consider

$$\begin{aligned} x+y+z&=6 \\ 2x-y+z&=3 \\ x+2y-z&=2. \end{aligned}$$

First, eliminate the same variable from two pairs of equations. Eliminate $x$ using the first equation:

$$R_2\leftarrow R_2-2R_1:\quad -3y-z=-9,$$

so

$$3y+z=9. \tag{1}$$

Next,

$$R_3\leftarrow R_3-R_1:\quad y-2z=-4. \tag{2}$$

The original three-variable system has now become a two-variable system. From equation $(2)$,

$$y=2z-4.$$

Substitute this into equation $(1)$:

$$3(2z-4)+z=9$$ $$7z=21\quad\Rightarrow\quad z=3.$$

Therefore, $y=2$. Substituting into $x+y+z=6$ gives $x=1$. Hence,

$$(x,y,z)=(1,2,3).$$

Check: $1+2+3=6$, $2(1)-2+3=3$, and $$. Therefore, the ordered triple satisfies all three original equations, not only the two reduced equations.

The final reduced equations also reveal the number of solutions:

A common misconception is that each equation should be solved separately. Instead, every elimination step must combine equations while preserving the common solution set.

Exam Technique

Show the row operation or equation combination explicitly before giving the answer. For a solve command, IB expects relevant algebraic stages, not only the final ordered triple. On Paper 2, even if a GDC solves the system, enter the system correctly and state the resulting solution; always substitute into the original equations to check it.