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1. Halle la ecuación de la recta que:
(a) Pasa por el (−2, 1) y es paralela a la recta 3x − 2y + 4 = 0.
(b) Es perpendicular a la recta 3x − 5y − 2 = 0 y pasa por el punto (−2, 1).

Let's solve the two parts of the problem step by step.

### Problem (a): Finding the Equation of a Line
**Given:**
- The line passes through the point \((-2, 1)\).
- It is parallel to the line \(3x - 2y + 4 = 0\).

**Step 1: Identify the slope of the given line.**

The equation of the line is in the standard form \(Ax + By + C = 0\). The slope \(m\) of a line in this form can be found using the formula:

\[
m = -\frac{A}{B}
\]

For the given line \(3x - 2y + 4 = 0\):
\[
A = 3, \quad B = -2
\]
\[
m = -\frac{3}{-2} = \frac{3}{2}
\]

Since the line we need to find is parallel to this line, it will have the same slope, \(m = \frac{3}{2}\).

**Step 2: Use the point-slope form to find the equation of the line.**

The point-slope form is given by:
\[
y - y_1 = m(x - x_1)
\]

Substituting \(m = \frac{3}{2}\) and the point \((-2, 1)\):
\[
y - 1 = \frac{3}{2}(x + 2)
\]

Distribute the slope:
\[
y - 1 = \frac{3}{2}x + 3
\]

Add 1 to both sides to get the equation in slope-intercept form:
\[
y = \frac{3}{2}x + 4
\]

The equation of the line is:
\[
\boxed{y = \frac{3}{2}x + 4}
\]

### Problem (b): Finding the Equation of a Line
**Given:**
- The line passes through the point \((-2, 1)\).
- It is perpendicular to the line \(3x - 5y - 2 = 0\).

**Step 1: Identify the slope of the given line.**

For the given line \(3x - 5y - 2 = 0\):
\[
A = 3, \quad B = -5
\]
\[
m = -\frac{3}{-5} = \frac{3}{5}
\]

The slope of a line perpendicular to this one will be the negative reciprocal of \(\frac{3}{5}\), which is:
\[
m_{\text{perpendicular}} = -\frac{5}{3}
\]

**Step 2: Use the point-slope form to find the equation of the line.**

Substitute \(m = -\frac{5}{3}\) and the point \((-2, 1)\):
\[
y - 1 = -\frac{5}{3}(x + 2)


Learn how to solve equations of lines that are parallel and perpendicular to given lines passing through the point (-2, 1). This problem involves using the point-slope form of a line and understanding the slopes of parallel and perpendicular lines. Suitable for high school students.

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