Solve each system in Exercises 5–18. 3x+2y−3z=−2, 2x−5y+2z=−2, 4x−3y+4z= 10

Verified step by step guidance

1. Write down the system of equations clearly: 1) $\mathrm{3x}+2y-3z=-2$, 2) $2x-5y+2z=-2$, 3) $4x-3y+4z=10$.
2. Choose a method to solve the system: substitution, elimination, or matrix methods. Here, elimination is effective. Start by eliminating one variable from two pairs of equations. For example, eliminate $z$ by combining equations 1 and 2, and then equations 1 and 3.
3. Multiply the equations as needed to align coefficients of $z$. For example, multiply equation 1 by 2 and equation 2 by 3 to make the coefficients of $z$ opposites: Equation 1 × 2: $6x\; +\; 4y\; -\; 6z\; =\; -4$, Equation 2 × 3: $6x\; -\; 15y\; +\; 6z\; =\; -6$.
4. Add the modified equations to eliminate $z$: $(6x\; +\; 4y\; -\; 6z)\; +\; (6x\; -\; 15y\; +\; 6z)\; =\; -4\; +\; (-6)$, which simplifies to an equation in $x$ and $y$ only. Repeat a similar process with equations 1 and 3 to get a second equation in $x$ and $y$.
5. Solve the resulting two-variable system for $x$ and $y$. Once you find $x$ and $y$, substitute these values back into one of the original equations to solve for $z$.

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. Understanding how to perform row operations like row swapping, scaling, and addition is crucial for applying Gaussian elimination or Gauss-Jordan elimination.