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The problem presents a scenario where a new metro line is being planned in a city. The Cartesian plane represents the coordinates of the city's government office and two new metro stations. The government office is located at point \( (0, 0) \). One of the stations is to be built east of the office at \( (3, 0) \), and the other station will be built north of the office at \( (0, 2) \).

The task is to identify the correct graph representing the line that connects the two stations: \( (3, 0) \) and \( (0, 2) \).

### Step-by-step solution:
1. **Coordinates of the points**:
   - Station 1: \( (3, 0) \) (east of the office).
   - Station 2: \( (0, 2) \) (north of the office).

2. **Equation of the line**:
   To find the equation of the line connecting these two points, we first calculate the slope \( m \):
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 0}{0 - 3} = \frac{2}{-3} = -\frac{2}{3}
   \]
   
   Using the point-slope form \( y - y_1 = m(x - x_1) \), and substituting one of the points \( (3, 0) \) and the slope \( -\frac{2}{3} \):
   \[
   y - 0 = -\frac{2}{3}(x - 3)
   \]
   Simplifying:
   \[
   y = -\frac{2}{3}x + 2
   \]
   This is the equation of the line.

3. **Correct graph**:
   The line has a negative slope, meaning it will slope downward from left to right. The y-intercept is \( 2 \), and the x-intercept is \( 3 \). Based on the options:
   - The correct graph is **A**, as it matches the equation \( y = -\frac{2}{3}x + 2 \).

Would you like more details on this or any related concepts?

### Related questions:
1. How do you find the slope of a line between two points?
2. What is the significance of the y-intercept in a linear equation?
3. How would the graph change if the slope were positive?
4. How can you verify the correctness of a graph based on its equation?
5. How do you convert between different forms of linear equations?

**Tip**: When graphing a line, always check the slope direction and intercepts to verify accuracy!

Pretende-se construir uma nova linha de metrô em uma cidade. Seu traçado será retilíneo e haverá duas estações próximas à prefeitura que, em relação a um sistema de coordenadas cartesianas, está localizada no ponto (0,0). Uma das estações ficará a leste da prefeitura, no ponto de coordenadas (3,0), e a outra ficará ao norte, no ponto de coordenadas (0,2). A nova linha de metrô está corretamente representada no plano cartesiano pela figura:

This problem involves determining the correct Cartesian graph representation of a metro line connecting two stations based on given coordinates. The solution requires understanding of slope calculation, equation of a line, and graph interpretation, making it ideal for students in Grades 10-12.

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