Find the solution set of each linear system. Identify inconsistent systems and dependent equations.
[tex]\left\{\begin{array}{l}
x+4 y-z=-8 \
x+5 y+z=-10 \
x-6 y+5 z=12\end{array}\right.[/tex]
Answer (2)
Eliminate x from the second and third equations.
Solve the resulting 2x2 system for y and z .
Substitute the values of y and z back into the first original equation to solve for x .
The solution set is ( 0 , − 2 , 0 ) .
Explanation
Analyzing 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 ( x , y , z ) that satisfies all three equations. We also need to determine if the system is inconsistent or if there are any dependent equations.
Restating the Equations The given system of equations is:
{ x + 4 y − z = − 8 x + 5 y + z = − 10 x − 6 y + 5 z = 12
We will use the elimination method to solve this system.
Eliminating x First, we eliminate x from the second and third equations. Subtract the first equation from the second equation:
( x + 5 y + z ) − ( x + 4 y − z ) = − 10 − ( − 8 ) y + 2 z = − 2
Next, subtract the first equation from the third equation:
( x − 6 y + 5 z ) − ( x + 4 y − z ) = 12 − ( − 8 ) − 10 y + 6 z = 20 − 5 y + 3 z = 10
Now we have a system of two equations with two unknowns, y and z :
{ y + 2 z = − 2 − 5 y + 3 z = 10
Solving for y and z Multiply the first equation by 5 to eliminate y :
5 ( y + 2 z ) = 5 ( − 2 ) 5 y + 10 z = − 10
Add this to the second equation:
( 5 y + 10 z ) + ( − 5 y + 3 z ) = − 10 + 10 13 z = 0 z = 0
Now substitute z = 0 into the equation y + 2 z = − 2 :
y + 2 ( 0 ) = − 2 y = − 2
So we have y = − 2 and z = 0 .
Solving for x Substitute y = − 2 and z = 0 into the first original equation to solve for x :
x + 4 ( − 2 ) − 0 = − 8 x − 8 = − 8 x = 0
So we have x = 0 , y = − 2 , and z = 0 .
Checking the Solution Check the solution ( 0 , − 2 , 0 ) in all three original equations:
0 + 4 ( − 2 ) − 0 = − 8 , which is true.
0 + 5 ( − 2 ) + 0 = − 10 , which is true.
0 − 6 ( − 2 ) + 5 ( 0 ) = 12 , which is true.
Since the solution satisfies all three equations, the solution set is ( 0 , − 2 , 0 ) .
Final Answer The solution set of the given linear system is ( 0 , − 2 , 0 ) . The system is consistent and has a unique solution. There are no dependent equations.
Examples
Systems of equations are incredibly useful in various real-world applications. For instance, consider a scenario where a business is trying to optimize its production. They produce three different products, each requiring different amounts of labor, materials, and machine time. By setting up a system of equations, where each equation represents a constraint (e.g., total labor hours available, total material available, etc.), the business can solve for the optimal production quantities of each product that maximize profit while staying within the resource constraints. This ensures efficient resource allocation and informed decision-making.
The solution set of the given linear system is (0, -2, 0). The system is consistent and has a unique solution with no dependent equations. All three equations are satisfied by this solution.
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