what is the equation of the line that is perpendicular to and has the same $y$-intercept as the given line?

$y = \frac{1}{5}x + 1$
$y = \frac{1}{5}x + 5$
$y = 5x + 1$
$y = 5x + 5$

Question

what is the equation of the line that is perpendicular to and has the same $y$-intercept as the given line?

$y = \frac{1}{5}x + 1$
$y = \frac{1}{5}x + 5$
$y = 5x + 1$
$y = 5x + 5$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Perpendicular Lines", "Slope-Intercept Form" ], "new_concepts": [], "current_concepts": [ "Perpendicular Lines", "Slope-Intercept Form", "Linear Equations" ] } ## Find the slope and y-intercept of the given line $$ \begin{aligned} &\text{Points: } (0, 1) \text{ and } (5, 0)\ &m_{\text{given}} = \frac{0 - 1}{5 - 0} = -\frac{1}{5}\ &b_{\text{given}} = 1 \end{aligned} $$ ## Determine the perpendicular slope and y-intercept $$ \begin{aligned} &m_{\perp} = -\frac{1}{m_{\text{given}}} = -\frac{1}{-\frac{1}{5}} = 5\ &b_{\perp} = b_{\text{given}} = 1 \end{aligned} $$ ## Write the equation of the perpendicular line $$ y = 5x + 1 $$

(A) $y = \frac{1}{5}x + 1$

(B) $y = \frac{1}{5}x + 5$

(C) $y = 5x + 1$

(D) $y = 5x + 5$

{ "elements": [ { "type": "point", "params": [[0, 1]], "properties": { "name": "(0, 1)", "size": 4, "color": "#583C87", "withLabel": true } }, { "type": "point", "params": [[5, 0]], "properties": { "name": "(5, 0)", "size": 4, "color": "#583C87", "withLabel": true } }, { "type": "line", "params": [[0, 1], [5, 0]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2 } }, { "type": "line", "params": [[0, 1], [1, 6]], "properties": { "strokeColor": "#F2557F", "strokeWidth": 2, "dash": 2 } } ] } { "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Perpendicular Lines" ] }

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QUESTION IMAGE

Question

Explanation:

Find the slope and y-intercept of the given line

Determine the perpendicular slope and y-intercept

Write the equation of the perpendicular line

Answer: