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. Step 1: Write down the system of equations: $4x-0y+2z=11$, $x+2y-z=-1$, $2x+2y-3z=-1$.
2. Step 2: Use the first equation to express $z$ in terms of $x$: $2z=11-4x$, so $z=\frac{11-4x}{2}$.
3. Step 3: Substitute $z=\frac{11-4x}{2}$ into the second equation: $x+2y-\frac{11-4x}{2}=-1$.
4. Step 4: Simplify the equation from Step 3 to find a relationship between $x$ and $y$.
5. Step 5: Substitute $z=\frac{11-4x}{2}$ into the third equation and solve for $x$ and $y$ using the equations from Steps 3 and 4.

Key Concepts

Systems of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The solution to the system is the set of values that satisfy all equations simultaneously. Methods to solve these systems include substitution, elimination, and matrix operations.

Gaussian Elimination

Gaussian elimination is a method used to solve systems of linear equations by transforming the system into an upper triangular form. This involves using row operations to simplify the equations, making it easier to back-substitute and find the values of the variables. It is particularly useful for larger systems.

Matrix Representation

Linear equations can be represented in matrix form, where coefficients of the variables are organized into a matrix, and constants are placed in a vector. This representation allows for the application of matrix operations, such as finding the inverse or using determinants, to solve the system efficiently.