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The problem provides parametric equations for a line \( r \) in the form:

\[
\begin{cases}
x = 2 - t \\
y = 1 + 4t \\
t \in \mathbb{R}
\end{cases}
\]

Let's break down and address each question:

### a) Do the points \( P(1, 5) \) and \( Q(3, -2) \) belong to \( r \)?
To check if these points belong to the line, we need to find whether there exists a value of \( t \) such that the parametric equations yield these points.

1. **Point \( P(1, 5) \):**
   - From the equation \( x = 2 - t \), set \( x = 1 \):
     \[
     1 = 2 - t \quad \Rightarrow \quad t = 1.
     \]
   - Check if \( y = 5 \) holds when \( t = 1 \):
     \[
     y = 1 + 4(1) = 5.
     \]
   So, \( P(1, 5) \) belongs to the line \( r \).

2. **Point \( Q(3, -2) \):**
   - From the equation \( x = 2 - t \), set \( x = 3 \):
     \[
     3 = 2 - t \quad \Rightarrow \quad t = -1.
     \]
   - Check if \( y = -2 \) holds when \( t = -1 \):
     \[
     y = 1 + 4(-1) = 1 - 4 = -3.
     \]
   Since \( y \neq -2 \), the point \( Q(3, -2) \) does **not** belong to the line \( r \).

### b) For what value of \( t \) is the point \( (-2, 17) \) on the line?

We solve for

Sean {x = 2 - t, y = 1 + 4t, t ∈ R}, las ecuaciones paramétricas de una recta r. a) ¿Los puntos P(1,5) y Q(3,-2) pertenecen a r? b) ¿Para qué valor del parámetro t se obtiene el punto (-2,17)? c) ¿Para qué valores de t se obtienen los puntos del segmento determinado por las intersecciones de la recta con los ejes coordenados? d) Calcule el área del triángulo que forma la recta con los ejes coordenados. e) Escriba otras ecuaciones paramétricas de la misma recta.

Explore parametric equations of the line x = 2 - t, y = 1 + 4t, and solve problems involving intersections with coordinate axes, checking if points belong to the line, and calculating the area of the triangle formed. Ideal for high school students in Grades 10-12.

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