Examkin

Soru

Zorluk: OrtaParallel and Perpendicular Lines

On a map of a city, Oak Street is represented by a straight line with the equation 3x−4y=123x - 4y = 12. A new road, Pine Street, is planned to be perpendicular to Oak Street and will pass through a park located at the coordinates (2,−1)(2, -1). Which of the following is an equation that represents Pine Street?

  1. A
    3x+4y=23x + 4y = 2
  2. B
    4x−3y=114x - 3y = 11
  3. 4x+3y=54x + 3y = 5Cevap
  4. D
    4x+3y=74x + 3y = 7
  5. E
    4x+3y=−94x + 3y = -9

Cevap

4x+3y=54x + 3y = 5
The slope of Oak Street is 34\frac{3}{4}. The perpendicular line representing Pine Street must have a slope that is the negative reciprocal of 34\frac{3}{4}, which is −43-\frac{4}{3}. Using the point-slope form with the point (2,−1)(2, -1), we get y−(−1)=−43(x−2)y - (-1) = -\frac{4}{3}(x - 2). Simplifying this yields y+1=−43x+83y + 1 = -\frac{4}{3}x + \frac{8}{3}, which becomes y=−43x+53y = -\frac{4}{3}x + \frac{5}{3}. Multiplying the entire equation by 3 and moving the xx term to the left side gives the standard form equation 4x+3y=54x + 3y = 5.

Adım Adım Çözüm

1
Find the slope of Oak Street by converting its equation to slope-intercept form (y=mx+by = mx + b).
The equation 3x−4y=123x - 4y = 12 becomes −4y=−3x+12-4y = -3x + 12, which simplifies to y=34x−3y = \frac{3}{4}x - 3. The slope of Oak Street is 34\frac{3}{4}.
To find the slope of a perpendicular line, we must first determine the slope of the original line.
2
Determine the perpendicular slope of Pine Street by taking the negative reciprocal of Oak Street's slope.
The negative reciprocal of 34\frac{3}{4} is −43-\frac{4}{3}.
Perpendicular lines have slopes that are negative reciprocals of each other.
3
Use the point-slope formula y−y1=m(x−x1)y - y_1 = m(x - x_1) with the point (2,−1)(2, -1) and the perpendicular slope −43-\frac{4}{3} to find the equation of Pine Street.
Substituting the values gives y−(−1)=−43(x−2)y - (-1) = -\frac{4}{3}(x - 2), which simplifies to y+1=−43x+83y + 1 = -\frac{4}{3}x + \frac{8}{3}.
The point-slope formula allows us to write the equation of a line given its slope and a point it passes through.
4
Convert the equation to standard form (Ax+By=CAx + By = C).
Subtract 1 from both sides: y=−43x+83−33⇒y=−43x+53y = -\frac{4}{3}x + \frac{8}{3} - \frac{3}{3} \Rightarrow y = -\frac{4}{3}x + \frac{5}{3}. Multiply the entire equation by 3: 3y=−4x+53y = -4x + 5. Add 4x4x to both sides: 4x+3y=54x + 3y = 5.
Converting to standard form matches the format of the options provided in the question.

Anahtar Kavram

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other (m1⋅m2=−1m_1 \cdot m_2 = -1). Once the perpendicular slope is found, the point-slope formula can be used to write the equation of the line passing through a given point.
Bu soruyu puanla

Konular

Parallel and Perpendicular Lines

Bu konudan daha fazla soru