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. Step 1: Write the system of equations in matrix form as an augmented matrix.
2. Step 2: Use the Gaussian elimination method to transform the augmented matrix into row-echelon form.
3. Step 3: Continue with the Gaussian elimination to achieve reduced row-echelon form.
4. Step 4: Once in reduced row-echelon form, interpret the matrix to find the values of x, y, and z.
5. Step 5: Verify the solution by substituting the values of x, y, and z back into the original equations to ensure they satisfy all three equations.

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, each providing a way to find the intersection point(s) of the equations.

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 performing row operations on the augmented matrix of the system, allowing for back substitution to find the values of the variables. It is a systematic approach that simplifies the solving process, especially for larger systems.

Matrix Representation

Matrix representation is a way to express a system of linear equations in a compact form using matrices. Each equation can be represented as a row in an augmented matrix, where the coefficients of the variables form the left part and the constants form the right part. This representation facilitates the application of various algebraic techniques, such as Gaussian elimination, to find solutions efficiently.