11.2 Systems of Linear Equations: Three Variables
3 min read•june 24, 2024
Systems of linear equations with three variables expand our problem-solving toolkit. These systems involve three equations and three unknowns, allowing us to model more complex real-world scenarios. We'll explore methods like elimination and substitution to solve these systems.
Understanding the consistency of three-equation systems is crucial. We'll learn to distinguish between consistent systems with unique or , and inconsistent systems with no solutions. This knowledge helps us interpret results and apply them to practical situations.
Solving Systems of Linear Equations with Three Variables
Solving three-variable linear systems
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- Multiply equations by appropriate constants eliminates one variable at a time (, , or )
- Add or subtract resulting equations obtains an equation with two variables
- Repeat process eliminates another variable, resulting in equation with one variable
- Solve for remaining variable and substitute back finds values of other variables
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- Solve one equation for one variable in terms of other two (express in terms of and )
- Substitute expression for solved variable into other two equations
- Solve resulting system of two equations with two variables using substitution or elimination
- Substitute values of two variables back into expression for third variable finds its value
Consistency of three-equation systems
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- Has at least one solution ( or infinitely many solutions)
- Equations represent planes that intersect at a point (one solution) or a line (infinite solutions)
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- Has (equations have no common point of intersection)
- Equations represent that do not intersect
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- Has infinite solutions (equations represent same plane or planes that intersect along a line)
- Equations are multiples of each other or one equation can be derived from others
Interpreting solutions for three-equation systems
- One solution (consistent and independent)
- Solution is an satisfies all three equations simultaneously
- Graphically, solution represents point of intersection of three planes
- No solution (inconsistent)
- System has no solution, equations have no common point of intersection
- Graphically, planes represented by equations are parallel and do not intersect
- Infinite solutions (consistent and dependent)
- System has infinite number of solutions, represented by a line or a plane
- Graphically, planes represented by equations coincide or intersect along a line
- Express solution using or as of variables (, where is a parameter)
- One variable may be a , allowing it to take any value while others are expressed in terms of it
Matrix Representation and Row Reduction
- : Organize coefficients of variables and constants into a matrix form
- : Transform the through row operations
- Leading coefficients (pivots) are 1
- Each leading 1 is in a column to the right of the leading 1 in the row above it
- : Further simplification of echelon form
- Each column containing a leading 1 has zeros in all other entries
- Simplifies solving for variables and identifying free variables
Applying Systems of Linear Equations with Three Variables
Solving real-world problems with three-variable linear systems
- Identify variables and quantities they represent in problem (let = number of apples, = number of bananas, = number of oranges)
- Set up system of three linear equations based on given information and relationships between variables
- Solve system using elimination, substitution, or other appropriate methods
- Interpret solution in context of original problem, ensuring values make sense in given situation (negative numbers of fruit are not possible)
Key Terms to Review (31)
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.
Back-Substitution: Back-substitution is a technique used to solve systems of linear equations, particularly those with three variables. It involves substituting the values of the variables found in earlier steps back into the original equations to determine the final values of the remaining variables.
Coefficient matrix: A coefficient matrix is a rectangular array that contains only the coefficients of the variables in a system of linear equations. It is used to facilitate methods such as Gaussian Elimination and finding matrix inverses.
Coefficient Matrix: A coefficient matrix, also known as the coefficient array, is a matrix that contains the coefficients of the variables in a system of linear equations. It is a crucial component in the analysis and solution of systems of linear equations, as it provides a compact and organized representation of the coefficients that define the relationships between the variables.
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.
Echelon Form: Echelon form is a way of representing a system of linear equations in matrix form, where the matrix is in a specific triangular arrangement. This form is particularly useful for analyzing and solving systems of linear equations, especially those with three or more variables.
Elimination Method: The elimination method, also known as the method of elimination, is a technique used to solve systems of linear equations by systematically eliminating variables to find the unique solution. This method is applicable in the context of various topics, including parametric equations, systems of linear equations in two and three variables, and systems of nonlinear equations and inequalities.
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.
Infinite Solutions: Infinite solutions refers to a situation where a system of equations has an unlimited number of solutions that satisfy all the equations in the system. This concept is particularly relevant in the context of systems of linear equations with three variables and systems of nonlinear equations and inequalities with two variables.
Intersecting Planes: Intersecting planes are two or more planes in three-dimensional space that share a common line or point. The intersection of these planes creates a line or point that represents the shared space between them.
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 Combination: A linear combination is the sum of a set of vectors, each multiplied by a corresponding scalar (numerical) coefficient. It represents a way of combining multiple vectors into a single vector by applying specific weights or coefficients to each vector.
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.
Ordered Triple: An ordered triple is a set of three numbers or values that are arranged in a specific order, typically denoted as (x, y, z). Ordered triples are used to represent points in a three-dimensional coordinate system and are an essential concept in the study of systems of linear equations with three variables.
Parallel Planes: Parallel planes are two or more planes in three-dimensional space that do not intersect and maintain a constant distance between them. They are an important concept in the study of systems of linear equations with three variables, as the equations representing these planes can be used to solve for the variables.
Parametric Equations: Parametric equations are a way of representing the coordinates of a point as functions of a parameter, typically denoted by the variable 't'. This allows for the description of curves and shapes that cannot be easily represented using traditional Cartesian coordinates.
Pivot Element: The pivot element is a key concept in the context of systems of linear equations with three variables. It refers to the leading, non-zero coefficient that is used to perform row operations and eliminate variables during the process of solving the system of equations.
Reduced Row Echelon Form: Reduced row echelon form is a special type of matrix representation where the matrix has been transformed to have a leading 1 in each row and 0s below it. This form is particularly useful in solving systems of linear equations, as it allows for the efficient identification of the solutions.
Row Reduction: Row reduction is a technique used to solve systems of linear equations by transforming the coefficient matrix into an equivalent matrix in row echelon form. This process simplifies the system and allows for the determination of the solutions to the equations.
Substitution method: The substitution method is a technique for solving systems of equations by substituting one equation into another. This transforms the system into a single-variable equation that can be solved more easily.
Substitution Method: The substitution method is a technique used to solve systems of linear equations, systems of nonlinear equations, and other types of equations by substituting one variable in terms of another. This method involves isolating a variable in one equation and then substituting that expression into the other equation(s) to solve for the remaining variable(s).
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.
X₁, x₂, x₃: In the context of systems of linear equations with three variables, x₁, x₂, and x₃ represent the three unknown quantities that need to be solved for. These variables are the solutions to the system of equations, which can be found using various methods such as substitution, elimination, or matrix methods.