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Let's go through the problems one by one. I'll start by solving the first one.

### Problem 1:
We are asked to find positive values of \( a \) and \( b \) in the equation of the line \( ax + by = 6 \) such that the line passes through the point \( P(3,1) \) and forms a triangle of area 6 with the coordinate axes.

1. **Substitute the point \( P(3, 1) \) into the line equation**:
   \[
   a(3) + b(1) = 6
   \]
   This simplifies to:
   \[
   3a + b = 6 \quad \text{(Equation 1)}
   \]

2. **The line intersects the x-axis and y-axis**:
   - At the x-axis (\( y = 0 \)): The x-intercept is \( \left( \frac{6}{a}, 0 \right) \).
   - At the y-axis (\( x = 0 \)): The y-intercept is \( \left( 0, \frac{6}{b} \right) \).

   These intercepts form a right triangle with the coordinate axes. The area of a triangle is given by:
   \[
   \text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
   \]
   Here, the base is \( \frac{6}{a} \) and the height is \( \frac{6}{b} \). The area is 6, so:
   \[
   \frac{1}{2} \times \frac{6}{a} \times \frac{6}{b} = 6
   \]
   Simplifying:
   \[
   \frac{36}{2ab} = 6 \quad \Rightarrow \quad \frac{36}{ab} = 6 \quad \Rightarrow \quad ab = 6 \quad \text{(Equation 2)}
   \]

3. **Solve the system of equations**:
   - From Equation 1: \( 3a + b = 6 \)
   - From Equation 2: \( ab = 6 \)

I'll solve this system next.It seems there are no positive real solutions for the system of equations as currently formulated. This suggests that there might be a mistake or additional constraint to consider.

Let’s review the equations:
1. \( 3a + b = 6 \)
2. \( ab = 6 \)

To avoid overcomplicating things, it may be useful to attempt solving this manually or try alternative methods. Would you like to move on to the next question or refine this one further?

1. (Valor: 2.0) Calcule a e b positivos na equação da reta ax + by = 6 de modo que ela passe pelo ponto P(3,1) e forme com os eixos coordenados um triângulo de área igual a 6.

Learn how to calculate the positive values of a and b in the line equation ax + by = 6 that passes through the point P(3,1) and forms a triangle with an area of 6 with the coordinate axes. This involves key geometry concepts and solving a system of equations.

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