Math Problem Statement

Test the consistency of the following system and solve if they are consistent: 3x - y - 2z = 2, 2y - z = -1, 3x - 5y = 3.

Solution

Let's test the consistency of the system of equations and solve it if it's consistent. The system is:

3x - y - 2z &= 2 \quad \text{(1)} \\ 2y - z &= -1 \quad \text{(2)} \\ 3x - 5y &= 3 \quad \text{(3)} \end{aligned}$$ ### Step 1: Solve for $$z$$ from Equation (2) From Equation (2): $$2y - z = -1 \implies z = 2y + 1 \quad \text{(4)}$$ ### Step 2: Substitute $$z = 2y + 1$$ into Equation (1) Substitute $$z = 2y + 1$$ from (4) into (1): $$3x - y - 2(2y + 1) = 2$$ $$3x - y - 4y - 2 = 2$$ $$3x - 5y - 2 = 2 \implies 3x - 5y = 4 \quad \text{(5)}$$ ### Step 3: Compare with Equation (3) Notice that Equation (5) is the same as Equation (3): $$3x - 5y = 3$$ and $$3x - 5y = 4$$ These two equations contradict each other. One equation states $$3x - 5y = 3$$ while the other states $$3x - 5y = 4$$. Therefore, **the system is inconsistent** and has no solution. ### Conclusion: The system of equations is inconsistent, so there is **no solution**. --- Would you like more details or have any questions? Here are 5 related questions you might explore: 1. How do we test consistency using matrices for a system of linear equations? 2. What are the methods to solve consistent systems of linear equations (e.g., substitution, elimination)? 3. What is the geometric interpretation of inconsistent systems? 4. Can inconsistent systems ever have a partial solution? 5. How do rank and the augmented matrix help in determining consistency? **Tip:** Inconsistent systems typically arise when two or more equations represent parallel lines in 2D, or planes in 3D, that do not intersect.

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Math Problem Analysis

Mathematical Concepts

Linear Systems
Algebra
Inconsistent Systems

Formulas

Substitution method: solve one equation for one variable and substitute in others
System consistency: compare the resulting equations after substitutions

Theorems

Consistency of systems of linear equations

Suitable Grade Level

Grades 10-12

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