Solve each system in Exercises 5–18. $\begin{cases}2x + y = 2 \\x + y - z = 4 \\3x + 2y + z = 0\end{cases}$

Verified step by step guidance

1. Write down the system of equations clearly: \[2x + y = 2\] \[x + y - z = 4\] \[3x + 2y + z = 0\]
2. From the first equation, express \(y\) in terms of \(x\): \[y = 2 - 2x\]
3. Substitute the expression for \(y\) into the second and third equations to eliminate \(y\): Second equation becomes: \[x + (2 - 2x) - z = 4\] Third equation becomes: \[3x + 2(2 - 2x) + z = 0\]
4. Simplify both equations to get two equations in terms of \(x\) and \(z\): For the second equation: \[x + 2 - 2x - z = 4\] For the third equation: \[3x + 4 - 4x + z = 0\]
5. Solve the simplified system of two equations with two variables (\(x\) and \(z\)) using substitution or elimination, then back-substitute to find \(y\).

Key Concepts

Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. Understanding how to interpret and set up these systems is essential for solving them.

Methods for Solving Systems (Substitution, Elimination, and Matrix Methods)

Common methods to solve systems include substitution, elimination, and using matrices (such as Gaussian elimination). These techniques help reduce the system to simpler forms, making it easier to find the values of variables that satisfy all equations.

Three-Variable Systems

When a system has three variables, it typically involves three equations. Solving such systems requires careful manipulation to eliminate variables step-by-step, often reducing the system to two variables and then one, to find the unique solution or determine if none or infinitely many exist.