Lesson 11.2: Systems of Linear Equations: Three Variables

New Concept

We're extending our skills to solve systems with three variables. You'll learn to find the unique solution $(x, y, z)$ where three planes intersect, and how to identify systems with infinite solutions or no solution at all.

What’s next

Next up, you'll tackle worked examples using elimination and back-substitution. Then, test your skills on practice cards and interactive challenge problems.

Property

In order to solve systems of equations in three variables, known as three-by-three systems, the primary tool we will be using is called Gaussian elimination. The goal is to eliminate one variable at a time to achieve upper triangular form. A system in upper triangular form looks like the following:

The third equation can be solved for $z$, and then we back-substitute to find $y$ and $x$. To write the system in upper triangular form, we can perform the following operations:

1. Interchange the order of any two equations.
2. Multiply both sides of an equation by a nonzero constant.
3. Add a nonzero multiple of one equation to another equation.

The solution set to a three-by-three system is an ordered triple ${(x, y, z)}$.

Examples

• To solve the system $x+y+z=6$, $2x-y+z=3$, and $x+y-z=0$, first add equation (1) and (2) to get $3x+2z=9$. Add equation (1) and (3) to get $2x=6$, so $x=3$. Substitute $x=3$ into $3x+2z=9$ to get $z=0$. Finally, substitute $x=3$ and $z=0$ into $x+y+z=6$ to find $y=3$. The solution is $(3, 0, 3)$.

• Find the solution for the system $x-y+2z=7$, $y+z=5$, and $3z=9$. Start with the last equation to find $z=3$. Back-substitute $z=3$ into the second equation to get $y+3=5$, so $y=2$. Finally, substitute $y=2$ and $z=3$ into the first equation: $x-2+2(3)=7$, which gives $x=3$. The solution is $(3, 2, 3)$.

Property

• Systems that have a single solution are those which, after elimination, result in a solution set consisting of an ordered triple ${(x, y, z)}$. Graphically, the ordered triple defines a point that is the intersection of three planes in space.
• Systems that have an infinite number of solutions are those which, after elimination, result in an expression that is always true, such as $0 = 0$. Graphically, an infinite number of solutions represents a line or coincident plane that serves as the intersection of three planes in space.
• Systems that have no solution are those that, after elimination, result in a statement that is a contradiction, such as $3 = 0$. Graphically, a system with no solution is represented by three planes with no point in common.

Examples

• A system with a single solution, like $x+y+z=3$, $2y=2$, and $3z=6$, solves to the unique ordered triple $(1, 1, 2)$. This means the three planes intersect at a single point.

• Consider the system $x+y+z=2$ and $2x+2y+2z=4$. Multiplying the first equation by -2 and adding it to the second results in $0=0$. This identity indicates there are infinite solutions.

Property

An inconsistent system of equations in three variables does not have a solution that satisfies all three equations. The process of elimination will result in a false statement, such as $3=7$ or some other contradiction. Graphically, the three planes do not intersect at a single point.

Examples

• Consider the system $x+y+z=5$, $2x+2y+2z=12$, and $x-y-z=1$. If we multiply the first equation by $-2$ and add it to the second, we get $( -2x-2y-2z ) + ( 2x+2y+2z ) = -10+12$, which simplifies to $0=2$. This is a contradiction, so there is no solution.

• Given the equations $x-2y+z=4$ and $-x+2y-z=1$, adding them together immediately yields the contradiction $0=5$. The system is inconsistent.

Property

A dependent system of equations has an infinite number of solutions. This happens when the elimination process yields an identity, such as $0=0$. When a system is dependent, we can find general expressions for the solutions. By adding equations and simplifying, we can express two of the variables in terms of the third (e.g., find $y$ and $z$ in terms of $x$). The general solution is an ordered triple like $(x, \frac{5}{2}x, \frac{3}{2}x)$, where $x$ can be any real number.

Examples

• In the system $x+y+z=4$, $2x+2y+2z=8$, and $x-y=1$, multiplying the first equation by $-2$ and adding it to the second results in $0=0$. This identity proves the system is dependent and has infinite solutions.

• Continuing with the system above, we express the solution in terms of one variable. From $x-y=1$, we get $y=x-1$. Substituting this into the first equation gives $x+(x-1)+z=4$, so $2x-1+z=4$, which means $z=5-2x$. The general solution is $(x, x-1, 5-2x)$.