what is the y-intercept of the line perpendicular to the line $y = \frac{3}{4}x + 3$ that includes the point $(3, 1)$?

* $-\frac{5}{4}$
* $3$
* $-\frac{13}{4}$
* $5$

Question

what is the y-intercept of the line perpendicular to the line $y = \frac{3}{4}x + 3$ that includes the point $(3, 1)$?

* $-\frac{5}{4}$
* $3$
* $-\frac{13}{4}$
* $5$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Perpendicular Lines", "Point-Slope Form" ], "new_concepts": [], "current_concepts": [ "Perpendicular Lines", "Point-Slope Form", "Linear Functions" ] } ## Determine the perpendicular slope $$ \begin{aligned} m_1 &= \frac{3}{4} \ m_2 &= -\frac{1}{m_1} = -\frac{4}{3} \end{aligned} $$ ## Find the equation of the perpendicular line $$ \begin{aligned} y - y_1 &= m_2(x - x_1) \ y - 1 &= -\frac{4}{3}(x - 3) \ y - 1 &= -\frac{4}{3}x + 4 \ y &= -\frac{4}{3}x + 5 \end{aligned} $$ ## Identify the y-intercept $$ \begin{aligned} b &= 5 \end{aligned} $$

(A) $-\frac{5}{4}$

(B) 3

(C) $-\frac{13}{4}$

(D) 5

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Perpendicular Lines" ] }

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QUESTION IMAGE

Question

Explanation:

Determine the perpendicular slope

Find the equation of the perpendicular line

Identify the y-intercept

Answer: