Find the solution set of each linear system. Identify inconsistent systems and dependent equations.

$\left\{\begin{aligned} x+4 y-z & =21 \\ x+5 y+z & =23 \\ x-6 y+5 z & =-25 \end{aligned}\right.$

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 \{(, , )\}.
B. The system is consistent and dependent. The solution set is the set of all ordered triples (, z) )
C. The system is inconsistent.

Eliminate x from the second and third equations.
Solve the resulting system of two equations for y and z .
Substitute the values of y and z back into the first equation to solve for x .
The solution set is ( 4 , 4 , − 1 ) ​ .

Explanation

Understanding the Problem We are given a system of three linear equations with three unknowns: x , y , and z . Our goal is to find the solution set of this system and determine if the system is inconsistent or contains dependent equations. The given equations are:


x + 4 y − z = 21
x + 5 y + z = 23
x − 6 y + 5 z = − 25


Eliminating x We can use the method of elimination or substitution to solve the system of equations. Let's start by eliminating x from equations (2) and (3). Subtract equation (1) from equation (2):

( x + 5 y + z ) − ( x + 4 y − z ) = 23 − 21 y + 2 z = 2 (4)
Now, subtract equation (1) from equation (3):
( x − 6 y + 5 z ) − ( x + 4 y − z ) = − 25 − 21 − 10 y + 6 z = − 46 (5)

Solving for z Now we have a system of two equations with two variables, y and z :

y + 2 z = 2 (4) − 10 y + 6 z = − 46 (5)
Multiply equation (4) by 10:
10 y + 20 z = 20 (6)
Add equation (5) and equation (6):
( 10 y + 20 z ) + ( − 10 y + 6 z ) = 20 + ( − 46 ) 26 z = − 26 z = − 1

Solving for y Substitute the value of z back into equation (4):

y + 2 ( − 1 ) = 2 y − 2 = 2 y = 4

Solving for x Now substitute the values of y and z back into equation (1):

x + 4 ( 4 ) − ( − 1 ) = 21 x + 16 + 1 = 21 x + 17 = 21 x = 4

Final Solution The solution set is ( x , y , z ) = ( 4 , 4 , − 1 ) . The system is consistent and has a unique solution.

Examples
Systems of linear equations are used in various fields, such as engineering, physics, economics, and computer science. For example, in structural engineering, these systems can be used to analyze the forces acting on a bridge or building. In economics, they can be used to model the supply and demand of goods in a market. Understanding how to solve these systems is crucial for making informed decisions and predictions in these fields. The ability to solve systems of equations allows engineers to design stable structures, economists to predict market trends, and computer scientists to develop efficient algorithms.

Answered by GinnyAnswer | 2025-07-08