y= 4x-x^2 sobre X y sobre la recta y= -3x+6

To solve the problem where you have \( y = \frac{4x - x^2}{x} \) and need to compare it with the line \( y = -3x + 6 \), let's follow these steps:

### Step 1: Simplify the Given Equation
The given equation is:

\[
y = \frac{4x - x^2}{x}
\]

Simplify this by dividing both terms in the numerator by \( x \):

\[
y = \frac{4x}{x} - \frac{x^2}{x}
\]

Simplifying further:

\[
y = 4 - x
\]

So, the equation simplifies to \( y = 4 - x \).

### Step 2: Intersection with the Line \( y = -3x + 6 \)
Now, we need to find the intersection of the simplified equation \( y = 4 - x \) with the line \( y = -3x + 6 \).

To do this, set the two equations equal to each other:

\[
4 - x = -3x + 6
\]

### Step 3: Solve for \( x \)
Solve the equation for \( x \):

1. Add \( 3x \) to both sides:

\[
4 + 2x = 6
\]

2. Subtract 4 from both sides:

\[
2x = 2
\]

3. Divide both sides by 2:

\[
x = 1
\]

### Step 4: Find \( y \)
Substitute \( x = 1 \) back into either equation to find \( y \). Using \( y = 4 - x \):

\[
y = 4 - 1 = 3
\]

### Step 5: Conclusion
The point of intersection of the curve \( y = 4 - x \) and the line \( y = -3x + 6 \) is at \( (1, 3) \).

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Would you like more details on this solution or have any questions? Here are five related questions to explore further:

1. How do you find the slope of a line from its equation?
2. What is the significance of the point of intersection between two lines or curves?
3. How do you solve quadratic equations graphically?
4. Can you find the intersection of two curves using a graphing method?
5. How does the shape of a parabola affect its intersections with a line?

**Tip:** Always double-check your work by substituting the solution back into the original equations.

Explore the intersection of the curve y = (4x - x^2) / x and the line y = -3x + 6. This solution includes simplification of algebraic expressions and solving systems of equations to find the point of intersection, suitable for students in Grades 8-10.

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