A system of equations is a set of two or more equations with the same variables.
A solution of a system of equations is a set of values for the variables that makes all the equations true.
In the previous section, we learned how to solve systems of equations with two variables using the elimination method. Now, we will extend this method to systems with three variables.
Solving Systems of Three Equations with Three Unknowns
A system of three equations with three unknowns looks like this:
$$\begin{aligned} a_1x + b_1y + c_1z &= d_1 \ a_2x + b_2y + c_2z &= d_2 \ a_3x + b_3y + c_3z &= d_3 \end{aligned}$$
The goal is to find the values of \$x\$, \$y\$, and \$z\$ that satisfy all three equations simultaneously.
The solution to a system of three equations with three unknowns is an ordered triple\$(x, y, z)\$.
The Elimination Method for Three Equations
The elimination method involves eliminating one variable at a time until you are left with a system of two equations with two unknowns, which can be solved using the methods from the previous section.
The key to the elimination method is to carefully choosewhich variable to eliminate first. Look for variables with matching coefficientsor multiplesof each other to make the elimination process easier.
The process of solving a system of three equations with three unknowns using the elimination method involves the following steps:
- Choose a variable to eliminate from two of the equations.
- Eliminate the same variable from a different pair of equations.
- Solve the resulting system of two equations with two unknowns.
- Substitute the solutions back into one of the original equations to find the value of the third variable.
How do we know that the elimination method will always work for systems of linear equations? Are there any limitations to this method?
