Solving Systems of Equations Using Gaussian Elimination

Systems of Equations and Inequalities

Solving Systems Using Gaussian Elimination

Systems of linear equations are a fundamental topic in College Algebra. One efficient method for solving such systems is Gaussian elimination, which transforms the system into an equivalent one that is easier to solve.

• System of Equations: A set of two or more equations with the same variables.

• Gaussian Elimination: A step-by-step process to convert a system of equations into row-echelon form using elementary row operations.

• Row-Echelon Form: A matrix form where each row has more leading zeros than the previous row, making back-substitution possible.

Example Problem

Solve the following system using Gaussian elimination:

• $-4x + 2y = 20$

• $x + y = -10$

Step-by-Step Solution

1. Write the system as an augmented matrix: $\begin{bmatrix} -4 & 2 & | & 20 \\ 1 & 1 & | & -10 \end{bmatrix}$

2. Use row operations to eliminate variables: - Multiply the second row by 4 and add to the first row to eliminate $x$ from the first equation. - Continue until the matrix is in row-echelon form.

3. Back-substitute to find the solution: - Solve for $y$ in the second equation, then substitute back to find $x$.

General Formulas

• Matrix Representation: $A\vec{x} = \vec{b}$, where $A$ is the coefficient matrix, $\vec{x}$ is the variable vector, and $\vec{b}$ is the constant vector.

• Elementary Row Operations:

  • Swap two rows

  • Multiply a row by a nonzero scalar

  • Add or subtract a multiple of one row to another row

Applications

• Solving real-world problems involving multiple constraints

• Used in engineering, economics, and computer science for optimization and modeling

Comparison Table: Gaussian vs. Gauss-Jordan Elimination

Method	Final Form	Solution Process
Gaussian Elimination	Row-Echelon Form	Back-substitution required
Gauss-Jordan Elimination	Reduced Row-Echelon Form	Solution is read directly from the matrix

Example Solution: For the system above, the solution set is the ordered pair $(x, y)$ that satisfies both equations.

Additional info: Gaussian elimination is a foundational technique for solving systems of equations and is closely related to matrix algebra, which is covered in College Algebra and Linear Algebra courses.