Solving Inconsistent Systems of Linear Equations: Example of -x

Math Problem Statement

Solve the system of equations: -x - 3y = -6 and x + 3y + 6 = 0.

You have shared an image showing a system of two linear equations labeled "System B":

-x - 3y &= -6 \\ x + 3y + 6 &= 0 \end{aligned}$$ Let's solve this system of equations using substitution or elimination. First, I'll simplify the second equation and proceed with solving the system. ### Step 1: Simplify the second equation $$x + 3y + 6 = 0 \quad \text{becomes} \quad x + 3y = -6$$ So now we have: $$\begin{aligned} -x - 3y &= -6 \\ x + 3y &= -6 \end{aligned}$$ ### Step 2: Add the two equations to eliminate variables $$(-x - 3y) + (x + 3y) = -6 + (-6)$$ This simplifies to: $$0 = -12$$ ### Conclusion: The result $$0 = -12$$ is a contradiction, which means that the system of equations is **inconsistent**. Therefore, **there is no solution** to this system. The lines represented by these equations are parallel and do not intersect. Would you like more details or have any questions? Here are 5 related questions: 1. How do we identify parallel lines in a system of equations? 2. Can a system of equations with a contradiction ever have solutions? 3. What is the graphical interpretation of inconsistent systems? 4. How does elimination help in solving systems of equations? 5. Are there different methods to solve inconsistent systems? **Tip**: Always check for contradictions when simplifying systems of equations, as it can quickly show if no solution exists!

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Algebra
Linear Equations
Systems of Equations
Inconsistency

Equation simplification
Addition of equations to eliminate variables

Parallel Lines Theorem

Grades 9-10

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