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

Verified step by step guidance

1. Write down the system of equations clearly: 1. $\mathrm{2x}+y=2$ 2. $x+y-z=4$ 3. $\mathrm{3x}+\mathrm{2y}+z=0$
2. Choose one equation to express one variable in terms of the others. For example, from the first equation, solve for $y$: $y=2-2x$
3. Substitute the expression for $y$ into the second and third equations to eliminate $y$. This will give you two equations with variables $x$ and $z$ only.
4. Simplify the two new equations and solve the resulting system of two equations with two variables ($x$ and $z$). You can use substitution or elimination methods here.
5. Once you find the values of $x$ and $z$, substitute them back into the expression for $y$ from step 2 to find the value of $y$.

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 that satisfy all equations.

Three-Variable Systems

Systems with three variables require solving three equations simultaneously. This often involves combining equations to eliminate variables step-by-step, reducing the system to two variables, then one, to find the unique solution or determine if none or infinitely many exist.