Instructor Guidance Example Week Three Discussion
Instructor Guidance Example Week Three Discussionparallel And Perpen
Identify the task of finding equations of lines parallel or perpendicular to given lines passing through specific points, using the point-slope form, and analyzing their intercepts. The assignment involves deriving equations, understanding slopes, and analyzing properties of lines in coordinate geometry.
Paper For Above instruction
The problem of determining equations for lines parallel or perpendicular to given lines constitutes a fundamental aspect of coordinate geometry, offering insights into the geometric relationships and algebraic representation of lines. This paper explores the methodology and calculations involved in deriving these equations, emphasizing the significance of slopes, point-slope form, and intercept analysis.
First, consider a line given by the equation \( y = -\frac{2}{3}x + 2 \). To find a line parallel to this, passing through the point \((-6, -3)\), one must recognize that parallel lines have identical slopes. Therefore, the slope of the new line will also be \( -\frac{2}{3} \). Using the point-slope form of a linear equation, \( y - y_1 = m(x - x_1) \), we substitute \( m = -\frac{2}{3} \) and the point \((-6, -3) \) to obtain:
\[
y - (-3) = -\frac{2}{3}(x - (-6))
\]
which simplifies to
\[
y + 3 = -\frac{2}{3}(x + 6)
\]
Expanding the right-hand side:
\[
y + 3 = -\frac{2}{3}x - 4
\]
Subtracting 3 from both sides:
\[
y = -\frac{2}{3}x - 7
\]
This is the equation of the line parallel to the original, passing through the specified point.
Next, for a line perpendicular to the original \( y = -\frac{2}{3}x + 2 \), which has slope \( -\frac{2}{3} \), the perpendicular line's slope is the negative reciprocal, \( \frac{3}{2} \). Passing through the point \((0, 5)\), again using the point-slope form:
\[
y - 5 = \frac{3}{2}(x - 0)
\]
which simplifies directly to:
\[
y = \frac{3}{2}x + 5
\]
This line has a gentle upward slope, crossing the y-axis at 5, and illustrating its perpendicular relation to the original line.
An important aspect of analyzing these lines involves the intercepts. The x-intercept of the parallel line, when \( y=0 \):
\[
0 = -\frac{2}{3}x - 7 \Rightarrow x = -\frac{7 \times 3}{-2} = 10.5
\]
indicating that the parallel line crosses the x-axis at \( 10.5 \). Similarly, the y-intercept is at \(-7\). For the perpendicular line, with the equation \( y = \frac{3}{2}x + 5 \), the x-intercept occurs at:
\[
0 = \frac{3}{2}x + 5 \Rightarrow x = -\frac{5 \times 2}{3} \approx -3.33
\]
and the y-intercept at 5.
Understanding the properties of these lines, such as the direction of increase or decrease in y-values with respect to x, reinforces their geometric relationships. The parallel line signifies equal slopes, maintaining the same angle with the x-axis, while the perpendicular line's slope being the negative reciprocal indicates a 90-degree angular difference.
In conclusion, deriving equations for parallel and perpendicular lines involve recognizing the intrinsic relationship between slopes, applying the point-slope form, and analyzing intercepts to understand their positions relative to axes. These exercises underpin more advanced applications in analytic geometry, such as distance calculations, angle between lines, and coordinate transformations, forming essential skills for mastery in mathematical analysis and its applications.
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