Lesson 4: Solving Linear Systems

Property

A linear equation with three variables, where $a$, $b$, $c$, and $d$ are real numbers and $a$, $b$, and $c$ are not all 0, is of the form $ax + by + cz = d$. Every solution to the equation is an ordered triple, $(x, y, z)$, that makes the equation true.

Solutions of a system of equations are the values of the variables that make all the equations true. A solution is represented by an ordered triple $(x, y, z)$.

Examples

• Is $(2, -1, 3)$ a solution to the system $\begin{cases} x+y+z=4 \\ 2x-y-z=2 \\ 3x+2y+z=7 \end{cases}$? Yes, because substituting the values makes all three equations true: $2+(-1)+3=4$, $2(2)-(-1)-3=2$, and $3(2)+2(-1)+3=7$.

Property

A solution to an equation in three variables, such as $x + 2y - 3z = -4$ is an ordered triple of numbers that satisfies the equation.

A solution to a system of three linear equations in three variables is an ordered triple that satisfies each equation in the system.

An ordered triple $(x, y, z)$ can be represented geometrically as a point in space using a three-dimensional Cartesian coordinate system, as shown in the figure.

Property

The elimination method can be extended to solve $3 \times 3$ linear systems by systematically eliminating variables to reduce the system to a simpler form.

Steps for Solving a $3 \times 3$ Linear System: