5x-9y=16 5x-9y=36 Consider the given system of equations. How many (x,y) solutions does this system have?

Let's consider the given system of equations:

\[
\text{Equation 1: } 5x - 9y = 16
\]
\[
\text{Equation 2: } 5x - 9y = 36
\]

To determine the number of solutions, let's compare the two equations.

### Step 1: Subtract the equations
We subtract Equation 1 from Equation 2:

\[
(5x - 9y) - (5x - 9y) = 36 - 16
\]

Simplifying this:

\[
0 = 20
\]

### Step 2: Analyze the result
The statement \( 0 = 20 \) is clearly false. This indicates that the system of equations is **inconsistent**, meaning there is no solution where both equations hold true simultaneously.

### Conclusion
The system of equations has **no solution**. Since the two lines represented by the equations are parallel but not identical, they do not intersect at any point.

Would you like a detailed explanation of any particular step? Here are five questions that might deepen your understanding:

1. How can you determine if two lines are parallel from their equations?
2. What does it mean for a system of equations to be inconsistent?
3. Can you explain what it means when the subtraction of two equations results in a false statement?
4. How do we determine the number of solutions for a system of linear equations in general?
5. What would the system look like if it had exactly one solution?

**Tip:** When subtracting one equation from another in a system, a false statement (like \( 0 = 20 \)) usually indicates the system is inconsistent, meaning no solutions exist.

Explore the solution to the system of linear equations 5x - 9y = 16 and 5x - 9y = 36. Understand why this system has no solutions, analyzing the concept of parallel lines and inconsistency in systems of equations. Suitable for Grades 9-12.

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