Find the solution set of the following linear system. Identify inconsistent systems and dependent equations.
[tex]\left\{\begin{aligned} x-3 y-z & = -9 \\ -x+8 y-4 z & = 24 \\ 2 x-15 y+7 z & = -45 \end{aligned}\right.[/tex]
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
A. The system is consistent and the solution set is {[tex]\([/tex] $\square$, $\square$, $\square$ )}.
(Simplify your answers.)
B. The system is consistent and dependent. The solution set is the set of all ordered triples {([tex]$\square$[/tex], [tex]$\square$[/tex], z) }, where
(Type an expression using z as the variable. Simplify your answer.)
C. The system is inconsistent.
Answer (2)
Write the augmented matrix for the system.
Perform row operations to reduce the matrix to row-echelon form.
Express x and y in terms of z.
The solution set is the set of all ordered triples ( 4 z , z + 3 , z ) , where z is any real number. ( 4 z , z + 3 , z )
Explanation
Analyzing the System We are given the following system of linear equations:
{ x − 3 y − z = − 9 − x + 8 y − 4 z = 24 2 x − 15 y + 7 z = − 45
Our goal is to find the solution set for this system. We will use Gaussian elimination to solve the system.
Forming the Augmented Matrix First, we write the augmented matrix for the system:
[ 1 − 3 − 1 ∣ − 9 − 1 8 − 4 ∣ 24 2 − 15 7 ∣ − 45 ]
We will perform row operations to reduce the matrix to row-echelon form.
Performing Row Operations We perform the following row operations:
R 2 = R 2 + R 1 :
[ 1 − 3 − 1 ∣ − 9 0 5 − 5 ∣ 15 2 − 15 7 ∣ − 45 ]
R 3 = R 3 − 2 R 1 :
[ 1 − 3 − 1 ∣ − 9 0 5 − 5 ∣ 15 0 − 9 9 ∣ − 27 ]
R 2 = 5 1 R 2 :
[ 1 − 3 − 1 ∣ − 9 0 1 − 1 ∣ 3 0 − 9 9 ∣ − 27 ]
R 3 = R 3 + 9 R 2 :
[ 1 − 3 − 1 ∣ − 9 0 1 − 1 ∣ 3 0 0 0 ∣ 0 ]
R 1 = R 1 + 3 R 2 :
[ 1 0 − 4 ∣ 0 0 1 − 1 ∣ 3 0 0 0 ∣ 0 ]
The matrix is now in reduced row-echelon form.
Finding the Solution Set From the reduced row-echelon form, we have the following system of equations:
{ x − 4 z = 0 y − z = 3
Solving for x and y in terms of z , we get:
x = 4 z y = z + 3
Thus, the solution set is the set of all ordered triples ( 4 z , z + 3 , z ) , where z is any real number.
Final Answer The system is consistent and dependent. The solution set is the set of all ordered triples ( 4 z , z + 3 , z ) , where z is any real number.
Examples
Systems of linear equations are used in various fields such as engineering, economics, and computer science. For example, in electrical engineering, Kirchhoff's laws can be modeled as a system of linear equations to analyze the currents and voltages in a circuit. In economics, supply and demand models can be represented as systems of linear equations to determine equilibrium prices and quantities. In computer graphics, systems of linear equations are used to perform transformations such as scaling, rotation, and translation of objects.
The solution set for the given system of equations is of the form (4z, z+3, z) where z is any real number. The system is consistent and dependent. Hence, the correct choice is B.
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