Systems of Equations With Three Variables

Two unknowns, two equations; three unknowns, three equations. That’s the pattern, and it’s why problems about three kinds of ticket, three investments or three numbers lead to a system of equations with three variables. You might expect a whole new technique. There isn’t one. You use the elimination method to knock out one variable, which turns the problem into a two-variable system, and you finish it exactly as before. In British textbooks this is often called solving three simultaneous equations with three unknowns.

What Is a System With Three Variables?

It’s three linear equations that share the same three variables, usually , and :

A solution is an ordered triple that makes all three equations true at once. Checking a triple works just like checking a pair: substitute into every equation. For the system

the triple gives , and . All three are true, so is the solution.

What the Solutions Look Like: Planes

With two variables, each equation is a line. With three, each equation is a plane, a flat sheet in three-dimensional space. Three planes can meet in three essentially different ways:

“No solution” can happen without any planes being parallel. Three planes can intersect in pairs and form a triangular tunnel, like the sides of a tent, with no single point on all three.

How to Solve a System of Equations With 3 Variables

1. Label the equations (1), (2) and (3), and choose one variable to eliminate. Pick the one that’s easiest to cancel.
2. Eliminate it from one pair, say (1) and (2), to get a new equation (4) in the other two variables.
3. Eliminate the same variable from a different pair, say (1) and (3) or (2) and (3), to get equation (5).
4. Solve the 2 × 2 system formed by (4) and (5).
5. Back-substitute both values into any original equation to find the third variable.
6. Check the ordered triple in all three original equations.

Step 3 is where most errors happen. You must eliminate the same variable both times, otherwise the two new equations still contain three variables between them.

Example 1: Solving by Elimination

Solve the system:

Step 1 — eliminate from (1) and (3). The -terms are opposites, so add:

Step 2 — eliminate from (2) and (3). Again the -terms are opposites:

Step 3 — solve (4) and (5). From (5), . Substitute into (4):

so .

Step 4 — back-substitute into (1):

Step 5 — check (2) and (3):

Answer:

Example 2: When Variables Are Already Isolated

Solve , and .

Two equations already express a variable in terms of , so the substitution method is quicker here.

Step 1 — substitute both into the first equation:

Step 2 — find the others:

Answer:

Triangular Form and Back-Substitution

Some systems arrive already “staircased”: the last equation has one variable, the one above has two, and the top one has three. This is called triangular form, and you solve it from the bottom up.

Turning any system into triangular form is exactly what Gaussian elimination does. That’s the matrix version of this lesson, and it’s how calculators solve systems with three, four or a hundred variables.

Special Cases: No Solution or Infinitely Many

No solution. Solve , and . Subtract the first equation from the second:

That’s false, so the system has no solution: the first two planes are parallel.

Infinitely many. Solve , and . Adding the first two equations gives , which is the third equation. Subtracting that from the third leaves

The third equation adds no new information. Subtracting the second equation from the first gives , so , and then . Every triple is a solution, which is a whole line of them.

Worked Example: A Three-Variable Word Problem

A theater sold tickets for . Child tickets cost , adult tickets , and senior tickets . Twice as many adult tickets were sold as child tickets. How many of each were sold?

Step 1 — define the variables: child, adult and senior tickets.

Step 2 — write three equations, one for each fact:

Step 3 — substitute into the count equation and solve for :

Step 4 — substitute both into the money equation:

so .

Step 5 — find the others:

Step 6 — check the money:

The theater sold 20 child, 40 adult and 40 senior tickets.

Common Mistakes to Avoid

• Eliminating different variables from the two pairs. If (4) has no but (5) has no , you still have three unknowns between them. Eliminate the same variable twice.
• Reusing the same pair. Combining (1) and (2) twice in different ways gives two equations that carry the same information. Use two different pairs.
• Forgetting the third variable. After solving the 2 × 2 system, you still need to back-substitute to get the last value.
• Checking only one equation. A triple can satisfy two equations and fail the third. Check all three.
• Losing a sign when subtracting. Subtracting changes the sign of every term: .

Practice Problems

Work each one before opening the answer.

Problem 1. Is a solution of , and ?

Problem 2. Solve the triangular system , , .

Problem 3. Solve , and .

Problem 4. Solve , and .

Problem 5. Solve , and .

Problem 6. Three numbers add up to . The middle number is more than the smallest, and the largest is twice the smallest. Find the numbers.

Writing three equations over and over, with all those ‘s, ‘s and ‘s, gets tedious. The next step is to strip the variables away and work only with the numbers, arranged in a grid called an augmented matrix.