Solve each system in Exercises 5–18. $\begin{cases}x + y = -4 \\y - z = 1 \\2x + y + 3z = -21\end{cases}$

Verified step by step guidance

1. Write down the system of equations clearly: \[x + y = -4\] \[y - z = 1\] \[2x + y + 3z = -21\]
2. From the first equation, express one variable in terms of the other. For example, solve for \[y\]: \[y = -4 - x\]
3. Substitute the expression for \[y\] into the second equation to relate \[x\] and \[z\]: \[(-4 - x) - z = 1\]
4. Simplify the equation from step 3 to express \[z\] in terms of \[x\]: \[z = -5 - x\]
5. Substitute the expressions for \[y\] and \[z\] from steps 2 and 4 into the third equation: \[2x + (-4 - x) + 3(-5 - x) = -21\], then simplify and solve for \[x\].

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. Such systems can be solved using substitution, elimination, or matrix methods.

Substitution and Elimination Methods

Substitution involves solving one equation for a variable and substituting that expression into other equations. Elimination involves adding or subtracting equations to eliminate a variable, simplifying the system. Both methods help reduce the system to fewer variables for easier solving.

Three-Variable Systems

When a system has three variables, it typically requires solving three equations simultaneously. Understanding how to manipulate and combine equations to isolate variables is essential. Solutions can be unique, infinite, or nonexistent depending on the system's consistency.