Question

look at the system of inequalities.
$y \leq 3x + 7$
$y \geq x - 3$
$y \leq 1$
the solution set is the triangular region where all the inequalities are true.
what are the vertices of that triangular region?
$(\square, \square)$
$(\square, \square)$
$(\square, \square)$

Explanation:

Step1: Encontrar intersección de \( y = 3x + 7 \) y \( y = 1 \)

Sustituir \( y = 1 \) en \( y = 3x + 7 \):
\( 1 = 3x + 7 \)
\( 3x = 1 - 7 = -6 \)
\( x = \frac{-6}{3} = -2 \)
Punto: \( (-2, 1) \)

Step2: Encontrar intersección de \( y = x - 3 \) y \( y = 1 \)

Sustituir \( y = 1 \) en \( y = x - 3 \):
\( 1 = x - 3 \)
\( x = 1 + 3 = 4 \)
Punto: \( (4, 1) \)

Step3: Encontrar intersección de \( y = 3x + 7 \) y \( y = x - 3 \)

Igualar las ecuaciones:
\( 3x + 7 = x - 3 \)
\( 3x - x = -3 - 7 \)
\( 2x = -10 \)
\( x = -5 \)
Sustituir \( x = -5 \) en \( y = x - 3 \):
\( y = -5 - 3 = -8 \)
Punto: \( (-5, -8) \)

Answer:

\( (-2, 1) \)
\( (4, 1) \)
\( (-5, -8) \)