Understanding Inconsistent Systems of Linear Equations

Math Problem Statement

solve the system of linear equations: x+y=8,x+y=12

Let's solve the system of linear equations:

\text{Equation 1:} & \quad x + y = 8 \quad \text{(1)} \\ \text{Equation 2:} & \quad x + y = 12 \quad \text{(2)} \end{align*}$$ ### Step 1: Analyze the System The equations can be directly compared: - Equation (1) states that the sum of $$x$$ and $$y$$ is 8. - Equation (2) states that the sum of $$x$$ and $$y$$ is 12. ### Step 2: Identify Any Contradictions If $$x + y$$ is equal to 8 in the first equation and equal to 12 in the second, this presents a contradiction because the same expression $$x + y$$ cannot equal two different values simultaneously. ### Conclusion Since the system presents a contradiction, there are no values of $$x$$ and $$y$$ that can satisfy both equations simultaneously. Therefore, **the system has no solution**. This type of system is called **inconsistent**. Would you like more details, or do you have any questions? --- Here are 5 related questions for further practice: 1. Determine if the system $$2x + 3y = 5$$ and $$2x + 3y = 10$$ has a solution. 2. Solve the system: $$3x - y = 7$$ and $$3x - y = 7$$. 3. Find the values of $$x$$ and $$y$$ if $$x - y = 4$$ and $$x - y = 6$$. 4. Determine whether the system $$5x + 2y = 11$$ and $$5x + 2y = 15$$ has one solution, no solution, or infinitely many solutions. 5. Solve for $$x$$ and $$y$$ in the system $$x + 2y = 6$$ and $$x + 2y = 6$$. **Tip:** Always check if two equations in a system are identical or contradictory; this helps quickly determine if the system has no solution or infinitely many solutions.

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Linear Equations
System of Equations
Inconsistency

Grades 7-9

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Solving Inconsistent Systems of Equations: x + 2y = 10, x + 2y = 8

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Solving System of Linear Equations: 2x - 2y = 1 and y - x = 2

Solving Inconsistent Systems of Linear Equations: Example of -x - 3y = -6 and x + 3y + 6 = 0