Solve each system in Exercises 5–18. $\begin{cases}3x + 2y - 3z = -2 \\2x - 5y + 2z = -2 \\4x - 3y + 4z = 10\end{cases}$

Verified step by step guidance

1. Write down the system of equations clearly: \[3x + 2y - 3z = -2\] \[2x - 5y + 2z = -2\] \[4x - 3y + 4z = 10\]
2. Choose a method to solve the system: substitution, elimination, or matrix methods. Here, elimination or substitution are common choices for three variables.
3. Use elimination or substitution to reduce the system from three equations with three variables to two equations with two variables. For example, eliminate one variable (like \(z\)) by combining pairs of equations.
4. Solve the resulting two-variable system for two variables (say \(x\) and \(y\)) using substitution or elimination.
5. Substitute the values of \(x\) and \(y\) back into one of the original equations to solve for \(z\), completing the solution for \((x, y, z)\).

Key Concepts

Systems of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of 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 transform the system into simpler forms to find the values of variables efficiently.

Consistency and Types of Solutions

Systems can have one unique solution, infinitely many solutions, or no solution. Recognizing the system's consistency helps determine the nature of the solution set and guides the solving process.