Systems of Linear Equations: Three Variables

Learning Outcomes

By the end of this section, you will be able to:

• Solve systems of three equations in three variables.
• Identify inconsistent systems of equations containing three variables.
• Express the solution of a system of dependent equations containing three variables.

Figure 1. (credit: “Elembis,” Wikimedia Commons)

Try It

Solve the system of equations in three variables.

\begin{align}2x+y - 2z&=-1 \\ 3x - 3y-z&=5 \\ x - 2y+3z&=6 \end{align}

Inconsistent and Dependent Systems in Three Variables

Just as with systems of equations in two variables, we may come across an inconsistent system of equations in three variables, which means that it does not have a solution that satisfies all three equations. The equations could represent three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location. The process of elimination will result in a false statement, such as $3=7$ or some other contradiction.

Example 4: Solving an Inconsistent System of Three Equations in Three Variables

Solve the following system.

\begin{align}x - 3y+z&=4&& \left(1\right) \\ -x+2y - 5z&=3&& \left(2\right) \\ 5x - 13y+13z&=8&& \left(3\right) \end{align}

Try It

Solve the system of three equations in three variables.

\begin{align}x+y+z&=2 \\ y - 3z&=1 \\ 2x+y+5z&=0 \end{align}

Expressing the Solution of a System of Dependent Equations Containing Three Variables

We know from working with systems of equations in two variables that a dependent system of equations has an infinite number of solutions. The same is true for dependent systems of equations in three variables. An infinite number of solutions can result from several situations. The three planes could be the same, so that a solution to one equation will be the solution to the other two equations. All three equations could be different but they intersect on a line, which has infinite solutions. Or two of the equations could be the same and intersect the third on a line.

Example 5: Finding the Solution to a Dependent System of Equations

Find the solution to the given system of three equations in three variables.

\begin{align}2x+y - 3z&=0&& \left(1\right)\\ 4x+2y - 6z&=0&& \left(2\right)\\ x-y+z&=0&& \left(3\right)\end{align}

Does the generic solution to a dependent system always have to be written in terms of $x?$

No, you can write the generic solution in terms of any of the variables, but it is common to write it in terms of x and if needed $x$ and $y$.

Try It

Solve the following system.

\begin{align}x+y+z&=7 \\ 3x - 2y-z&=4 \\ x+6y+5z&=24 \end{align}

Key Concepts

• A solution set is an ordered triple $\left\{\left(x,y,z\right)\right\}$ that represents the intersection of three planes in space.
• A system of three equations in three variables can be solved by using a series of steps that forces a variable to be eliminated. The steps include interchanging the order of equations, multiplying both sides of an equation by a nonzero constant, and adding a nonzero multiple of one equation to another equation.
• Systems of three equations in three variables are useful for solving many different types of real-world problems.
• A system of equations in three variables is inconsistent if no solution exists. After performing elimination operations, the result is a contradiction.
• Systems of equations in three variables that are inconsistent could result from three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location.
• A system of equations in three variables is dependent if it has an infinite number of solutions. After performing elimination operations, the result is an identity.
• Systems of equations in three variables that are dependent could result from three identical planes, three planes intersecting at a line, or two identical planes that intersect the third on a line.

Glossary

solution set
the set of all ordered pairs or triples that satisfy all equations in a system of equations