Solve each system in Exercises 5–18. 4x−0y+2z=11, x+2y−z=−1, 2x+2y−3z=−1

Verified step by step guidance

1. Write down the system of equations clearly: 1) $\mathrm{4x}-\mathrm{0y}+\mathrm{2z}=11$, 2) $x+\mathrm{2y}-z=-1$, 3) $\mathrm{2x}+\mathrm{2y}-\mathrm{3z}=-1$.
2. Choose a method to solve the system: substitution, elimination, or matrix methods. Here, elimination or substitution can be effective. For example, start by expressing $x$ from the second equation: $x=-1-\mathrm{2y}+z$.
3. Substitute the expression for $x$ into the first and third equations to eliminate $x$. This will give you two equations in terms of $y$ and $z$ only.
4. Simplify the resulting two equations and solve the system of two equations with two variables ($y$ and $z$) using substitution or elimination.
5. Once you find the values of $y$ and $z$, substitute them back into the expression for $x$ to find the value of $x$. This will give you the solution triplet $(x,\; y,\; z)$.

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 transform the system into simpler forms to find the values of variables efficiently.

Three-Variable Systems and Their Solutions

Systems with three variables often require careful manipulation to reduce equations step-by-step. Solutions can be unique, infinite, or nonexistent, depending on the system's consistency and independence, which is determined by the relationships among the equations.