4.3 Q-13

Question Statement

Find an equation of the line passing through the point $(-4, -6)$ and perpendicular to a line with slope $\frac{-3}{2}$.

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Background and Explanation

To solve this problem, we need to:

1. Understand slopes of perpendicular lines: The slope of a line perpendicular to another is the negative reciprocal of the original line’s slope.
2. Point-Slope Form: The equation of a line can be written in point-slope form, $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope of the line.

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Solution

1. Given Data:

   • The slope of the original line is $m = \frac{-3}{2}$.
   • The point through which the new line passes is $(-4, -6)$.

2. Finding the slope of the perpendicular line:

   • The slope of the line perpendicular to the given line is the negative reciprocal of $\frac{-3}{2}$.
   • The negative reciprocal of $\frac{-3}{2}$ is $\frac{2}{3}$.

3. Using the point-slope form:

   • We can now use the point-slope form of the equation to find the equation of the perpendicular line.
   • The point-slope form is:
     $y - y_1 = m_1(x - x_1)$
     where $(x_1, y_1)$ is the given point $(-4, -6)$, and $m_1 = \frac{2}{3}$ is the slope.

4. Substitute the values:

   • Substitute $x_1 = -4$, $y_1 = -6$, and $m_1 = \frac{2}{3}$ into the point-slope form:
     $y - (-6) = \frac{2}{3}(x - (-4))$
     Simplifying the equation:
     $y + 6 = \frac{2}{3}(x + 4)$

5. Simplify the equation:

   • Multiply both sides by 3 to eliminate the fraction:
     $3(y + 6) = 2(x + 4)$
     Expand both sides:
     $3y + 18 = 2x + 8$
     Rearranging the equation:
     $2x - 3y - 10 = 0$

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Key Formulas or Methods Used

• Point-Slope Form:
  $y - y_1 = m(x - x_1)$
  Used to write the equation of a line when a point on the line and the slope are known.

• Slope of Perpendicular Lines:
  If two lines are perpendicular, the product of their slopes is $-1$. The slope of the line perpendicular to a given line is the negative reciprocal of the original line’s slope.

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Summary of Steps

1. Find the slope of the perpendicular line, which is the negative reciprocal of $\frac{-3}{2}$, i.e., $\frac{2}{3}$.
2. Apply the point-slope form using the given point $(-4, -6)$ and the perpendicular slope $\frac{2}{3}$.
3. Simplify the equation to get the final result: $2x - 3y - 10 = 0$.