The problem is in German and can be translated to:

"Determine the coordinates of the intersection point of the lines $g$ and $h$ analytically. The graph of $g$ has a slope $m = -\frac{1}{3}$ and intersects the y-axis at point $P(0|2)$. The graph of $h$ passes through the points $Q(0|-2)$ and $R(6|-2)$. Draw both lines in a coordinate system."

Let's solve this step-by-step:

1. Equation of Line $g$

Given:

• Slope $m = -\frac{1}{3}$
• y-intercept $P(0|2)$

The equation of line $g$ in slope-intercept form $y = mx + b$ is: $y = -\frac{1}{3}x + 2$

2. Equation of Line $h$

Given points:

• $Q(0, -2)$
• $R(6, -2)$

Since both points have the same y-coordinate, line $h$ is a horizontal line. The equation is: $y = -2$

3. Finding the Intersection Point

To find the intersection, set the equations equal to each other: $-\frac{1}{3}x + 2 = -2$

Solve for $x$: $-\frac{1}{3}x = -2 - 2$ $-\frac{1}{3}x = -4$ Multiply by $-3$: $x = 12$

Now, substitute $x = 12$ into the equation of either line to find $y$: $y = -2 \quad (\text{since it's the equation of line } h)$

4. Intersection Point

Thus, the coordinates of the intersection point are: $(12, -2)$

Would you like a detailed explanation of how we found the intersection, or have any further questions about this problem?

Follow-Up Questions

1. How would the intersection point change if the slope of line $g$ were different?
2. Can you determine the x-intercept of line $g$ using its equation?
3. What is the general process to find the intersection point of two lines?
4. How would you graph these lines in a coordinate system?
5. What happens if two lines are parallel in terms of intersection points?

Tip

When finding the intersection point of two lines, set their equations equal to each other and solve for one variable, then substitute back to find the other variable.