Solve each system in Exercises 5–18. 2x−4y+3z=17, x+2y−z=0, 4x−y−z=6

Verified step by step guidance

1. Write down the system of equations clearly: $\mathrm{2x}-4y+3z=17,x+2y-z=0,4x-y-z=6$.
2. Choose a variable to eliminate first. For example, solve the second equation for $x$ in terms of $y$ and $z$: $x=-2y+z$.
3. Substitute this expression for $x$ into the first and third equations to get two equations with only $y$ and $z$.
4. Simplify these two new equations and solve the resulting system of two equations with two variables 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 $x$.

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 of Equations

Common methods to solve systems include substitution, elimination, and matrix techniques such as Gaussian elimination. These methods transform the system into simpler forms to find the values of variables efficiently. Choosing the appropriate method depends on the system's complexity.

Matrix Representation and Row Operations

Systems of linear equations can be represented as augmented matrices, allowing the use of row operations to simplify and solve the system. Row operations include swapping rows, multiplying a row by a nonzero scalar, and adding multiples of one row to another, which help achieve row-echelon form for back-substitution.