Solve each system in Exercises 5–18. x+y=−4, y−z=1, 2x+y+3z=−21

Verified step by step guidance

1. Identify the system of equations: $x+y=-4$, $y-z=1$, and $2x+y+3z=-21$.
2. From the first equation, express $y$ in terms of $x$: $y=\; -x-\; 4$.
3. Substitute this expression for $y$ into the second and third equations to eliminate $y$ and obtain two equations with variables $x$ and $z$ only.
4. Solve the resulting two-equation system for $x$ and $z$ using substitution or elimination methods.
5. Once $x$ and $z$ are found, substitute back into the expression for $y$ to find its value, completing the solution of the system.

Key Concepts

Systems of Linear Equations

A system of linear equations consists of two or more linear equations with 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 help reduce the system to simpler forms, making it easier to find the values of variables. Choosing the appropriate method depends on the system's complexity.

Three-Variable Systems

Systems with three variables require solving for three unknowns, often involving three equations. This adds complexity compared to two-variable systems, requiring careful manipulation to isolate variables and ensure consistency across all equations.