4.3 Q-14

Question Statement

Find the equation of the line passing through the point $(11, -5)$ and parallel to a line with slope $-24$.

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Background and Explanation

To solve this problem, we need to:

1. Understand parallel lines: Parallel lines have the same slope. Therefore, the slope of the required line will be the same as the given 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 given line is $m = -24$.
   • The point through which the new line passes is $(11, -5)$.

2. Slope of the parallel line:

   • Since the lines are parallel, the slope of the new line will also be $m = -24$.

3. Using the point-slope form:

   • We will use the point-slope form of the equation to find the equation of the line:
     $y - y_1 = m(x - x_1)$
     where $(x_1, y_1) = (11, -5)$ and $m = -24$.

4. Substitute the values:

   • Substituting the values into the point-slope form:
     $y - (-5) = -24(x - 11)$
     Simplifying the equation:
     $y + 5 = -24(x - 11)$

5. Simplify the equation:

   • Expand the right-hand side:
     $y + 5 = -24x + 264$
   • Rearranging the equation to bring it into standard form:
     $24x + y - 259 = 0$

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Key Formulas or Methods Used

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

• Parallel Lines:
  Parallel lines have the same slope. Therefore, the slope of the required line is equal to the slope of the given line.

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Summary of Steps

1. Identify the slope of the parallel line, which is $-24$ (same as the given line).
2. Use the point-slope form with the point $(11, -5)$ and slope $-24$.
3. Simplify the equation to get the final result: $24x + y - 259 = 0$.