To solve this system of linear equations, let’s examine the three equations provided:

1. $-2x - 3y - 2z = -8$
2. $-4x - 4y = -19$
3. $2x + 3y - 2z = 20$

We can solve this system step by step.

Step 1: Simplify and analyze the system

From Equation 2, there is no $z$-term. It only involves $x$ and $y$. This will allow us to reduce the system step by step.

Step 2: Work on Equations 1 and 3

From the first and third equations, we can eliminate $z$ to express the equations only in terms of $x$ and $y$. Let’s multiply Equation 1 and Equation 3 appropriately to eliminate $z$: $\text{Multiply Equation 1 by } 1: \, -2x - 3y - 2z = -8$ $\text{Multiply Equation 3 by } 1: \, 2x + 3y - 2z = 20$

Add these two equations: $(-2x + 2x) + (-3y + 3y) + (-2z - 2z) = -8 + 20$ $0 = 12$

Step 3: Analyze the result

The equation $0 = 12$ is a contradiction, meaning that the system of equations is inconsistent. There are no solutions to this system because the equations represent planes that do not intersect at a common point.

Final Answer:

• The system is inconsistent.

Would you like me to go into more details or explore similar examples?

Follow-up Questions:

1. How do you identify inconsistent systems graphically?
2. What are dependent systems of equations, and how do they differ from inconsistent systems?
3. Can an inconsistent system have any practical applications?
4. How do elimination and substitution methods reveal inconsistencies in systems?
5. What is the geometric interpretation of each equation in this system?

Tip:

When solving systems of equations, always simplify or rewrite equations to isolate fewer variables to make inconsistencies or solutions easier to detect.