Lesson 29: Solving Systems of Equations in Three Variables

New Concept

The solution to a system of three equations in three variables is an ordered triple and is noted $(x, y, z)$.

What’s next

Next, you’ll master a 5-step process using elimination to find the solution, or ordered triple, for systems with three variables.

The solution to a system of three equations in three variables is an ordered triple and is noted $(x, y, z)$. A solution means that there is an ordered triple that satisfies all three equations.

For the solution $(1, -6, 9)$, the point must satisfy all three equations, like $x+y+z=4$, to be correct. It's the one spot where they all agree! For example, if you test the point $(2, 1, 3)$ in the system $x+y+z=6$, $2x-y+z=6$, and $x-y-z=-2$, it works for all three. So, $(2, 1, 3)$ is the solution.

Think of an ordered triple as the secret coordinates $(x, y, z)$ to a treasure chest! This single point is where three different flat planes, representing your equations, all meet. If your coordinates don't work for all three planes, you've found the wrong spot and there is no treasure for you.

Step 1: Eliminate one variable from two equations. Step 2: Eliminate the same variable using a different pair of equations. Step 3: Solve the new two-variable system. Step 4: Substitute back to find the third variable. Step 5: Check your solution in all three original equations.

Given $x+y+z=9$ and $2x+y-z=3$, add them to eliminate $z$ and get $3x+2y=12$. This is Step 1. Then, you might eliminate $z$ from another pair, like $x-y+z=5$ and $2x+y-z=3$ to get $3x=8$. After finding $x$ and $y$, plug them into an original equation like $x+y+z=9$ to find $z$.

Solving these systems is like a puzzle! First, you team up two equations to knock out a variable. Then, you do it again with another pair to get a simple system. Solve that, work backward to find the missing piece, and double-check to make sure you've cracked the code correctly. It’s a foolproof plan!

A system that has at least one solution is consistent. A system of simultaneous equations that have no common solution is inconsistent.

If elimination leads to a true statement like $0=0$, the system is consistent with infinite solutions. If elimination leads to a false statement like $0=10$, the system is inconsistent because parallel planes never intersect, such as in the equations $x+y+z=5$ and $2x+2y+2z=15$.

Think of it like planning with friends! A 'consistent' system is when everyone agrees on a meeting spot (one point) or a path to walk together (a line). An 'inconsistent' system is when your friends' plans are impossible to sync up—they're on different parallel paths and will never, ever meet. No solution means no fun!