Problem 37
Question
Use matrices to solve each system of equations. If the equations of a system are dependent or if a system is inconsistent, state this. $$ \left\\{\begin{array}{l} 2 x+3 y-z=-8 \\ x-y-z=-2 \\ -4 x+3 y+z=6 \end{array}\right. $$
Step-by-Step Solution
Verified Answer
The system has a unique solution. x = 1, y = -2, z = 5.
1Step 1: Isolate the variable
Rearrange the equation/inequality using algebraic operations to isolate the variable.
2Step 2: Find the solution
The system has a unique solution. x = 1, y = -2, z = 5..
Key Concepts
Understanding Systems of EquationsMatrices in Solving Systems of EquationsIdentifying Dependent and Inconsistent Systems
Understanding Systems of Equations
A system of equations is a collection of two or more equations with a common set of variables. The objective is to find values for these variables that satisfy all the equations at once.
There are several methods to solve systems of equations, including substitution, elimination, and matrix algebra. In real-world applications, systems of equations can model various scenarios, such as calculating budget constraints or predicting trends.
For example, in the given system of equations, we can imagine each equation representing a line in three-dimensional space. The solution, if it exists, corresponds to the point(s) where these lines intersect, revealing the values of \(x\), \(y\), and \(z\) that satisfy all equations.
There are several methods to solve systems of equations, including substitution, elimination, and matrix algebra. In real-world applications, systems of equations can model various scenarios, such as calculating budget constraints or predicting trends.
For example, in the given system of equations, we can imagine each equation representing a line in three-dimensional space. The solution, if it exists, corresponds to the point(s) where these lines intersect, revealing the values of \(x\), \(y\), and \(z\) that satisfy all equations.
Matrices in Solving Systems of Equations
Matrices are incredibly useful in solving systems of equations, particularly when dealing with three or more variables. By writing the system of equations in matrix form, we simplify complex calculations and make solving the system more systematic.
A matrix is essentially a rectangular array of numbers arranged in rows and columns. In this context, we form a coefficient matrix, a variable matrix, and a constant matrix. Using matrix notation helps to organize the information. This makes it easier to apply methods like row reduction or use software tools for solutions.
For our problem, the coefficient matrix \(A\), variable matrix \( \mathbf{x} \), and constant matrix \( \mathbf{b} \) are:
A matrix is essentially a rectangular array of numbers arranged in rows and columns. In this context, we form a coefficient matrix, a variable matrix, and a constant matrix. Using matrix notation helps to organize the information. This makes it easier to apply methods like row reduction or use software tools for solutions.
For our problem, the coefficient matrix \(A\), variable matrix \( \mathbf{x} \), and constant matrix \( \mathbf{b} \) are:
- \( A = \begin{pmatrix} 2 & 3 & -1 \ 1 & -1 & -1 \ -4 & 3 & 1 \end{pmatrix} \)
- \( \mathbf{x} = \begin{pmatrix} x \ y \ z \end{pmatrix} \)
- \( \mathbf{b} = \begin{pmatrix} -8 \ -2 \ 6 \end{pmatrix} \)
Identifying Dependent and Inconsistent Systems
When working with systems of equations, it's crucial to determine if the system is dependent or inconsistent.
A dependent system has infinitely many solutions. This occurs when the equations represent the same geometric object, like overlapping lines. Here, no single solution is unique, as infinite points make the equations true simultaneously.
On the other hand, an inconsistent system has no solutions. This situation arises when the equations represent parallel lines that never intersect. Since there's no point common to all equations, the system can't be solved.
In matrix terms, we often look at the row-echelon form or reduced row-echelon form of the matrix. If we find a row like \( [0 \ 0 \ 0 | \ c] \) with a non-zero \( c \), the system is inconsistent. Conversely, if the system can be reduced to a point where one equation is a multiple of another, it is dependent. This analysis helps identify the nature of the system before attempting solutions.
A dependent system has infinitely many solutions. This occurs when the equations represent the same geometric object, like overlapping lines. Here, no single solution is unique, as infinite points make the equations true simultaneously.
On the other hand, an inconsistent system has no solutions. This situation arises when the equations represent parallel lines that never intersect. Since there's no point common to all equations, the system can't be solved.
In matrix terms, we often look at the row-echelon form or reduced row-echelon form of the matrix. If we find a row like \( [0 \ 0 \ 0 | \ c] \) with a non-zero \( c \), the system is inconsistent. Conversely, if the system can be reduced to a point where one equation is a multiple of another, it is dependent. This analysis helps identify the nature of the system before attempting solutions.
Other exercises in this chapter
Problem 37
Determine whether each equation defines \(y\) to be a function of \(x .\) If it does not, find two ordered pairs where more than one value of \(y\) corresponds
View solution Problem 37
Manufacturing. \(\quad\) A manufacturer of automobile water pumps is considering retooling for one of two manufacturing processes, with monthly fixed costs and
View solution Problem 37
Evaluate each determinant. $$ \left|\begin{array}{lll} 1 & 2 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{array}\right| $$
View solution Problem 37
Use a graphing calculator to solve each system. Give all answers to the nearest hundredth. See Using Your Calculator: Solving Systems by Graphing. $$ \left\\{\b
View solution