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Let's go through each question in the image:

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**3) Conociendo las coordenadas de dos puntos del plano** \( P_1(x_1, y_1) \) **y** \( P_2(x_2, y_2) \), **obtenga la ecuación explícita de la recta que contiene a estos dos puntos (considere** \( x_1 \neq x_2 \)**).**

The equation of the line passing through two points can be found using the formula:

\[
y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)
\]

This is the point-slope form of a line equation, where the slope \(m\) is given by:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

You can then simplify this to get the explicit form of the line \(y = mx + b\).

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**4) Escriba la ecuación explícita de la recta que contiene a** \( A(2, 3) \) **y forma un ángulo de 120° con el eje** \( x \).**

To solve this, first, note that the slope of the line \( m \) is related to the angle \( \theta \) it forms with the positive direction of the x-axis through:

\[
m = \tan(\theta)
\]

Since the angle is 120°, we can calculate:

\[
m = \tan(120^\circ) = \tan(180^\circ - 60^\circ) = -\tan(60^\circ) = -\sqrt{3}
\]

Thus, the slope of the line is \( m = -\sqrt{3} \). Now, using the point-slope form of the line equation:

\[
y - 3 = -\sqrt{3}(x - 2)
\]

You can expand and simplify to get the explicit form of the line.

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**5) Halle la ecuación explícita de la recta que contiene a los puntos** \( P_1(2, -3) \) **y** \( P_2(1, 5) \).**

To find the equation of the line through these two points, use the same approach as in question 3:

1. Calculate the slope \( m \):

\[
m = \frac{5 - (-3)}{1 - 2} = \frac{5 + 3}{-1} = \frac{8}{-1} = -8
\]

2. Now, use the point-slope form of the equation. Using point \( P_1(2, -3) \):

\[
y - (-3) = -8(x - 2)
\]

Simplify this equation to get the explicit form.

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Do you want detailed step-by-step solutions for each of these questions?

Here are five additional related questions to consider:
1. How do you find the slope of a line when given two points?
2. What is the significance of the angle a line makes with the x-axis?
3. How does the point-slope form compare to the slope-intercept form of a line equation?
4. What would the equation of the line be if the angle was 45° instead of 120°?
5. How can you check if a point lies on a given line equation?

**Tip:** Always double-check calculations involving trigonometric values, especially when dealing with non-standard angles.

1. Conociendo las coordenadas de dos puntos del plano P1(x1, y1) y P2(x2, y2), obtenga la ecuación explícita de la recta que contiene a estos dos puntos (considere x1 ≠ x2). 2. Escriba la ecuación explícita de la recta que contiene a A(2,3) y forma un ángulo de 120° con el eje x. 3. Halle la ecuación explícita de la recta que contiene a los puntos P1(2, -3) y P2(1, 5).

Explore how to find the explicit equation of a line given two points, a point and an angle with the x-axis, or using the point-slope form. Learn how to calculate slopes and apply trigonometric principles for advanced high school mathematics.

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