Question:
Quantity A: The slope of a line parallel to \(4y + 18x = 13\).
Quantity B: The slope of a line perpendicular to \(6y - 16x = 15\).
Which of the following is true?
Quantity A: The slope of a line parallel to \(4y + 18x = 13\).
Quantity B: The slope of a line perpendicular to \(6y - 16x = 15\).
Which of the following is true?
Quantity B: The slope of a line perpendicular to \(6y - 16x = 15\).
Which of the following is true?
Show Hint
Parallel lines have equal slopes; perpendicular slopes are negative reciprocals \(m_1 m_2 = -1\).
Updated On: Oct 3, 2025
- The two quantities are equal.
- The relationship between the quantities cannot be determined from the information provided.
- Quantity B is larger.
- Quantity A is larger.
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The Correct Option is C
Solution and Explanation
Step 1: Slope for Quantity A (parallel to \(4y + 18x = 13\)).
Write in slope–intercept form: \(4y = -18x + 13 \Rightarrow y = \left(-\tfrac{18}{4}\right)x + \tfrac{13}{4} = \left(-\tfrac{9}{2}\right)x + \tfrac{13}{4}\).
So any line parallel has slope \(m_A = -\tfrac{9}{2}\).
Step 2: Slope for Quantity B (perpendicular to \(6y - 16x = 15\)).
\(6y = 16x + 15 \Rightarrow y = \left(\tfrac{16}{6}\right)x + \tfrac{15}{6} = \left(\tfrac{8}{3}\right)x + \tfrac{5}{2}\).
Slope of the given line is \(\tfrac{8}{3}\). The perpendicular slope is the negative reciprocal: \(m_B = -\tfrac{3}{8}\).
Step 3: Compare.
\(-\tfrac{9}{2} = -4.5\) and \(-\tfrac{3}{8} = -0.375\).
Since \(-0.375>-4.5\), \(\boxed{\text{Quantity B is larger}}\).
Write in slope–intercept form: \(4y = -18x + 13 \Rightarrow y = \left(-\tfrac{18}{4}\right)x + \tfrac{13}{4} = \left(-\tfrac{9}{2}\right)x + \tfrac{13}{4}\).
So any line parallel has slope \(m_A = -\tfrac{9}{2}\).
Step 2: Slope for Quantity B (perpendicular to \(6y - 16x = 15\)).
\(6y = 16x + 15 \Rightarrow y = \left(\tfrac{16}{6}\right)x + \tfrac{15}{6} = \left(\tfrac{8}{3}\right)x + \tfrac{5}{2}\).
Slope of the given line is \(\tfrac{8}{3}\). The perpendicular slope is the negative reciprocal: \(m_B = -\tfrac{3}{8}\).
Step 3: Compare.
\(-\tfrac{9}{2} = -4.5\) and \(-\tfrac{3}{8} = -0.375\).
Since \(-0.375>-4.5\), \(\boxed{\text{Quantity B is larger}}\).
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