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

Let's solve the system of linear equations:

\[
\begin{align*}
\text{Equation 1:} & \quad x + y = 8 \quad \text{(1)} \\
\text{Equation 2:} & \quad x + y = 12 \quad \text{(2)}
\end{align*}
\]

### 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?

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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.

Explore the concept of inconsistent systems of linear equations with an example: solve the system of equations x + y = 8 and x + y = 12. Learn why such systems have no solution and understand the key mathematical concepts involved. Suitable for Grades 7-9.

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