Find the solution set of the following linear system. Identify inconsistent systems and dependent equations.
[tex]\left\{\begin{aligned} x+y & =0 \\ y+3 z & =-4 \\ y+z & =5-x \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]\{(\square, \square, \square)\}[/tex]}.
B. The system is consistent and dependent. The solution set is the set of all ordered triples {[tex]\{\square\}[/tex]}.
C. The system is inconsistent.
Answer (2)
Rewrite the third equation: x + y + z = 5 .
Solve the first equation for x: x = − y .
Substitute x = − y into the third equation to find z = 5 .
Substitute z = 5 into the second equation to find y = − 19 , then find x = 19 . The solution is ( 19 , − 19 , 5 ) .
Explanation
Analyzing the Problem We are given a system of three linear equations with three variables x, y, and z. The goal is to find the solution set and determine if the system is inconsistent or contains dependent equations.
Listing the Equations The given equations are:
x + y = 0
y + 3 z = − 4
y + z = 5 − x
Rewriting the Third Equation First, we rewrite the third equation to have all variables on the left side:
x + y + z = 5
Summarizing the System Now we have the following system of equations:
x + y = 0
y + 3 z = − 4
x + y + z = 5
Solving for x From equation (1), we can express x in terms of y :
x = − y
Substituting x into Equation 3 Substitute x = − y into equation (3):
− y + y + z = 5
z = 5
Substituting z into Equation 2 Now that we have z = 5 , we can substitute it into equation (2):
y + 3 ( 5 ) = − 4
y + 15 = − 4
y = − 19
Finding x Now we can find x using x = − y :
x = − ( − 19 )
x = 19
Stating the Solution So the solution is x = 19 , y = − 19 , and z = 5 . We can write this as an ordered triple ( 19 , − 19 , 5 ) .
Verifying the Solution To verify the solution, we substitute the values into the original equations:
19 + ( − 19 ) = 0 (True)
− 19 + 3 ( 5 ) = − 19 + 15 = − 4 (True)
− 19 + 5 = 5 − 19 − 14 = − 14 (True)
Since all equations are satisfied, the solution is correct.
Final Answer The system is consistent, and the solution set is {( 19 , − 19 , 5 )} .
Examples
Systems of equations are used in various real-world applications, such as determining the optimal mix of products to maximize profit, balancing chemical equations, and analyzing electrical circuits. For example, a company might use a system of equations to determine how many units of each product to produce in order to maximize profit, given constraints on resources such as labor and materials. By solving the system of equations, the company can find the optimal production levels that satisfy the constraints and maximize profit.
The system of equations is consistent, with a solution set of {(19, -19, 5)}.
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