Solve each system in Exercises 5–18. $\begin{cases}4x - y + 2z = 11 \\x + 2y - z = -1 \\2x + 2y - 3z = -1\end{cases}$

Verified step by step guidance

1. Write down the system of equations clearly: \[4x - 0y + 2z = 11\] \[x + 2y - z = -1\] \[2x + 2y - 3z = -1\]
2. Since the first equation has no y term, focus on expressing one variable in terms of the others from one of the simpler equations. For example, from the second equation, solve for \(x\) in terms of \(y\) and \(z\): \[x = -1 - 2y + z\]
3. Substitute the expression for \(x\) from step 2 into the first and third equations to eliminate \(x\). This will give you two equations with only \(y\) and \(z\) as variables.
4. Solve the resulting two-variable system from step 3 using either substitution or elimination to find values for \(y\) and \(z\).
5. Once you have \(y\) and \(z\), substitute these values back into the expression for \(x\) from step 2 to find the value of \(x\). This completes the solution for the system.

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

Consistency and Types of Solutions

A system can have one unique solution, infinitely many solutions, or no solution. Recognizing the system's consistency helps determine the nature of the solution set, which is crucial when interpreting the results of solving the system.