11.6 Solving Systems with Gaussian Elimination
3 min read•june 24, 2024
is a powerful method for solving systems of linear equations. It transforms equations into augmented matrices, then uses to simplify them into . This process reveals solutions and helps classify systems as having unique, infinite, or no solutions.
Understanding is crucial for tackling complex linear systems. It provides a systematic approach to solving equations, making it an essential tool in algebra, engineering, and data analysis. Mastering this technique opens doors to more advanced mathematical concepts.
Gaussian Elimination
Augmented matrices from equations
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- Represents a system of linear equations in form
- Each row corresponds to an equation
- Entries in each row are coefficients (numbers multiplying variables) of variables in the equation
- Last column contains constants from right-hand side of equations
- Creating an from a system of equations:
- Write coefficients of each in same order for each equation
- Add constants from right-hand side as last column
- Separate coefficients and constants with a vertical line
- Creating a system of equations from an :
- Each row represents an equation
- Entries in each row (except last column) are coefficients of variables
- Last column contains constants from right-hand side of equations
Row operations for echelon form
- Row echelon form is a matrix format where:
- First non-zero entry in each row () is 1
- Pivot of each row is located to the right of pivot in row above
- All other entries in column below a pivot are zero
- Three types of transform matrix into row echelon form:
- Swap positions of two rows
- Multiply a row by a non-zero constant
- Add a multiple of one row to another row
- Transforming a matrix into row echelon form:
- Use row operations to make first non-zero entry in first column equal to 1 (pivot)
- Use row operations to make all entries below pivot in first column zero
- Repeat process for next column, starting with next row down
- Continue until matrix is in row echelon form
Interpreting row echelon form
- Row echelon form of augmented matrix determines solution of
- Types of solutions:
- : Single solution set
- Pivot in every column (except last) in row echelon form
- : Infinite number of solution sets
- At least one (column without pivot) in row echelon form
- : No solution set
- Row where all entries are zero except last column in row echelon form
- : Single solution set
Gaussian elimination for systems
- Method for solving systems of linear equations using row operations
- Transforms augmented matrix into row echelon form
- Steps for Gaussian elimination:
- Write system of equations as augmented matrix
- Use row operations to transform matrix into row echelon form
- Interpret row echelon form to determine solution of system
- Systems with two variables:
- Row echelon form has two rows
- Solution found by solving for one variable in terms of other (if unique solution)
- Systems with three variables:
- Row echelon form has three rows
- Solution found by solving for one variable in terms of others (if unique solution)
- If infinitely many solutions, express solution in terms of free variable(s)
Linear Systems and Matrices
- A linear system is a set of equations where each variable appears to the first power
- A matrix is a rectangular array of numbers used to represent a linear system
- Variables in a linear system are the unknowns to be solved for
Key Terms to Review (28)
~: The tilde symbol (~) is a mathematical operator used in various contexts, including systems of linear equations and Gaussian elimination. In the context of solving systems with Gaussian elimination, the tilde represents the transformation of the original system of equations into an equivalent system with a reduced row echelon form (RREF).
Augmented matrix: An augmented matrix is a matrix that represents a system of linear equations, including both the coefficients and the constants from the equations. It combines the coefficient matrix and the constant vector into one larger matrix for easier manipulation and solution.
Augmented Matrix: An augmented matrix is a special type of matrix that is used to represent a system of linear equations. It is formed by combining the coefficient matrix of the system with the column of constants on the right-hand side of the equations.
Binomial coefficient: A binomial coefficient is a coefficient of any of the terms in the expansion of a binomial raised to a power, typically written as $\binom{n}{k}$ or $C(n,k)$. It represents the number of ways to choose $k$ elements from a set of $n$ elements without regard to order.
Coefficient: A coefficient is a numerical factor that multiplies a variable in an algebraic expression. It represents the magnitude or strength of the relationship between the variable and the overall expression. Coefficients are essential in various mathematical contexts, including polynomial factorization, linear equations, quadratic equations, and the graphing of polynomial functions.
Consistent system: A consistent system is a set of equations that has at least one solution. In a graph, the lines representing the equations intersect at one or more points.
Consistent System: A consistent system is a set of linear equations that has at least one solution. In other words, a consistent system is one where the equations can be satisfied simultaneously, meaning there exists a set of values for the variables that make all the equations true.
Dependent system: A dependent system is a system of linear equations in which all equations represent the same line, resulting in infinitely many solutions. This occurs when the equations are scalar multiples of one another.
Dependent System: A dependent system, in the context of linear equations, refers to a system where the equations are linearly dependent, meaning that one equation can be expressed as a linear combination of the other equations. This implies that the system has an infinite number of solutions or no solution at all.
Dependent variable: The dependent variable is the output of a function, whose value depends on the input or independent variable. It is usually represented as $y$ in the equation $y = f(x)$.
Elementary Row Operations: Elementary row operations are fundamental transformations performed on the rows of a matrix to simplify the matrix and solve systems of linear equations. These operations preserve the solutions of the original system while making it easier to find the solution.
Free Variable: A free variable is a variable in a system of equations that can be assigned any value without affecting the validity of the solution. It is a variable that is not constrained by the equations in the system, allowing for flexibility in finding a solution.
Gaussian elimination: Gaussian elimination is a method for solving systems of linear equations. It transforms the system's augmented matrix into row-echelon form using row operations.
Gaussian Elimination: Gaussian elimination is a method for solving systems of linear equations by transforming the system into an equivalent one that is easier to solve. It involves a series of row operations on the augmented matrix of the system to obtain an upper triangular matrix, which can then be used to find the solution to the system.
Inconsistent system: An inconsistent system is a set of equations that has no solution. This typically occurs when the equations represent parallel lines that never intersect.
Inconsistent System: An inconsistent system is a system of linear equations that has no solution, meaning there is no set of values for the variables that satisfies all the equations simultaneously. This term is particularly relevant in the context of solving systems of linear equations in two or more variables, as well as the techniques of Gaussian elimination and Cramer's rule.
Infinitely many solutions: Infinitely many solutions refers to a situation in a system of equations where there are countless values that satisfy all the equations simultaneously. This typically occurs when the equations are dependent, meaning that at least one equation can be derived from another. In this context, it often indicates that the system represents the same line or plane in space, allowing for an unlimited number of intersection points.
Leading coefficient: The leading coefficient of a polynomial is the coefficient of the term with the highest degree. It plays a crucial role in determining the end behavior of the polynomial function.
Leading Coefficient: The leading coefficient of a polynomial is the numerical coefficient of the term with the highest degree. It is the first, or leading, coefficient in the standard form of a polynomial expression. The leading coefficient plays a crucial role in understanding and analyzing various topics in college algebra, including polynomials, quadratic equations, functions, and more.
Linear System: A linear system is a collection of linear equations that describe the relationship between multiple variables. These equations can be solved simultaneously to find the values of the variables that satisfy all the equations in the system.
Matrix: A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns, that can be used to represent and manipulate mathematical relationships and data. Matrices are fundamental tools in various areas of mathematics, including linear algebra, applied mathematics, and computer science.
No Solution: The term 'no solution' refers to a situation where a system of equations or inequalities has no values for the variables that satisfy all the equations or inequalities simultaneously. In other words, there is no set of values that can be assigned to the variables that make all the expressions in the system true.
Pivot: In the context of solving systems with Gaussian elimination, a pivot is the leading, non-zero entry in a row that is used as the basis for the elimination process. The pivot is the key element that allows the elimination method to systematically reduce the system of equations to an equivalent system in row echelon form.
Row Echelon Form: Row echelon form is a particular arrangement of the rows in a matrix or system of linear equations that simplifies the process of solving the system using techniques like Gaussian elimination. It is a crucial concept in the context of solving systems of linear equations.
Row operations: Row operations are procedures used to manipulate the rows of a matrix in order to solve systems of equations. They include row swapping, multiplying a row by a non-zero scalar, and adding or subtracting multiples of rows.
Row Operations: Row operations refer to the fundamental mathematical techniques used to manipulate the rows of a matrix or system of linear equations. These operations are essential for solving systems of equations using methods like Gaussian Elimination and matrix inverse.
Unique solution: A unique solution refers to a single, distinct answer to a system of equations where all variables can be solved explicitly, resulting in one point of intersection in a graph. This concept is essential in understanding how various systems behave, especially when analyzing the relationships between multiple variables, whether linear or nonlinear. Identifying a unique solution ensures that the system is consistent and that there is a clear and definitive outcome for the values of the variables involved.
Variable: A variable is a symbol or letter that represents an unknown or changeable value in a mathematical expression, equation, or function. It serves as a placeholder for a value that can vary or be assigned different values within a given context.