How many points of intersection would the following system have? Explain: y=2x-4 and 2y=4x+5

Let's analyze the system of equations:

1. **First equation:** \( y = 2x - 4 \)
2. **Second equation:** \( 2y = 4x + 5 \)

### Step 1: Simplify the second equation
We start by simplifying the second equation. Divide both sides of \( 2y = 4x + 5 \) by 2 to isolate \( y \):

\[
y = 2x + \frac{5}{2}
\]

Now, we have the system of equations as:
1. \( y = 2x - 4 \)
2. \( y = 2x + \frac{5}{2} \)

### Step 2: Set the two equations equal to each other
Since both equations represent \( y \), we can set them equal to each other to find the point(s) of intersection:

\[
2x - 4 = 2x + \frac{5}{2}
\]

### Step 3: Solve for \( x \)
Subtract \( 2x \) from both sides:

\[
-4 = \frac{5}{2}
\]

This results in a contradiction because \( -4 \) is not equal to \( \frac{5}{2} \). Therefore, there is no solution for \( x \).

### Conclusion:
Since there is no solution, the two lines are **parallel** and do not intersect. Therefore, the system has **no points of intersection**.

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Here are some related questions:
1. What does it mean for two lines to be parallel?
2. How do we classify systems of equations in terms of solutions?
3. How can you graphically identify parallel lines?
4. What are the conditions for two lines to intersect?
5. How can you determine if two lines are identical or overlap?

**Tip:** If two linear equations have the same slope but different y-intercepts, they are parallel and will never intersect.

Explore the method to determine the number of points of intersection for the linear equations y=2x-4 and 2y=4x+5. This problem delves into algebraic concepts such as systems of equations and the parallel lines theorem. Ideal for students in Grades 8-10.

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