15. Given the line 3𝑥 − 2𝑦 = 12, give a line parallel through (−5 , 1) and a line perpendicular through (2 , 4)

To find both the parallel and perpendicular lines through the given points, we first need to determine the slope of the original line \( 3x - 2y = 12 \).

### Step 1: Rewrite the equation in slope-intercept form ( \( y = mx + b \) )

Start with the given line:
\[
3x - 2y = 12
\]
Solve for \( y \) in terms of \( x \):
\[
-2y = -3x + 12 \quad \Rightarrow \quad y = \frac{3}{2}x - 6
\]
Thus, the slope of the original line is \( m = \frac{3}{2} \).

### Step 2: Find the equation of the parallel line
Lines that are parallel have the same slope. Therefore, the slope of the line parallel to \( y = \frac{3}{2}x - 6 \) is also \( \frac{3}{2} \).

We now need to find the equation of the line that passes through the point \( (-5, 1) \) and has slope \( m = \frac{3}{2} \). Using the point-slope form of a line:
\[
y - y_1 = m(x - x_1)
\]
Substitute \( m = \frac{3}{2} \), \( x_1 = -5 \), and \( y_1 = 1 \):
\[
y - 1 = \frac{3}{2}(x + 5)
\]
Simplify:
\[
y - 1 = \frac{3}{2}x + \frac{15}{2}
\]
\[
y = \frac{3}{2}x + \frac{15}{2} + 1
\]
\[
y = \frac{3}{2}x + \frac{17}{2}
\]
Thus, the equation of the parallel line is:
\[
y = \frac{3}{2}x + \frac{17}{2}
\]

### Step 3: Find the equation of the perpendicular line
Lines that are perpendicular have slopes that are negative reciprocals. The negative reciprocal of \( \frac{3}{2} \) is \( -\frac{2}{3} \).

We need to find the equation of the line passing through the point \( (2, 4) \) with slope \( m = -\frac{2}{3} \). Again, using the point-slope form:
\[
y - y_1 = m(x - x_1)
\]
Substitute \( m = -\frac{2}{3} \), \( x_1 = 2 \), and \( y_1 = 4 \):
\[
y - 4 = -\frac{2}{3}(x - 2)
\]
Simplify:
\[
y - 4 = -\frac{2}{3}x + \frac{4}{3}
\]
\[
y = -\frac{2}{3}x + \frac{4}{3} + 4
\]
Convert \( 4 \) to a fraction:
\[
y = -\frac{2}{3}x + \frac{4}{3} + \frac{12}{3}
\]
\[
y = -\frac{2}{3}x + \frac{16}{3}
\]
Thus, the equation of the perpendicular line is:
\[
y = -\frac{2}{3}x + \frac{16}{3}
\]

### Final Answer:
1. The equation of the **parallel line** through \( (-5, 1) \) is:
   \[
   y = \frac{3}{2}x + \frac{17}{2}
   \]
2. The equation of the **perpendicular line** through \( (2, 4) \) is:
   \[
   y = -\frac{2}{3}x + \frac{16}{3}
   \]

Would you like more details on any step?

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Here are 5 related questions you might want to explore:
1. How do you derive the point-slope form of a line?
2. How can we verify that two lines are perpendicular?
3. What happens to the graph when the slope of a line increases or decreases?
4. How can we determine if two lines are neither parallel nor perpendicular?
5. How do you find the intersection point of two lines?

**Tip:** The slope tells you the steepness of a line. Parallel lines have identical slopes, while perpendicular lines have slopes that are negative reciprocals of each other.

Given the line 3𝑥 − 2𝑦 = 12, give a line parallel through (−5, 1) and a line perpendicular through (2, 4).

Learn how to find a parallel line through the point (-5, 1) and a perpendicular line through (2, 4) for the equation 3x - 2y = 12. This algebra problem demonstrates key concepts like slope and the relationships between parallel and perpendicular lines, suitable for students in Grades 9-11.

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