perpendicular y= 1/4 x+7

{ "query": { "display": "perpendicular $$y=\\frac{1}{4}x+7$$", "symbolab_question": "PRE_CALC#perpendicular y=\\frac{1}{4}x+7" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Line Equations", "subTopic": "Perpendicular", "default": "y=-4x+b", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "General line, perpendicular to $$y=\\frac{1}{4}x+7:{\\quad}y=-4x+b$$", "steps": [ { "type": "interim", "title": "Find the slope of $$y=\\frac{1}{4}x+7:{\\quad}m=\\frac{1}{4}$$", "input": "y=\\frac{1}{4}x+7", "steps": [ { "type": "step", "primary": "For a line equation for the form of $$\\mathbf{y=mx+b}$$, the slope is $$\\mathbf{m}$$", "result": "m=\\frac{1}{4}" } ], "meta": { "interimType": "Slope Equation Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMGyxztx1DIZ4Y9QoeLjXWS3gvQsnK15dYEA+LzNRkamakCNQBrqjAVqHinrSnCzdJeKao7hW0voz3VcTgg+4km8lPniFtXc+FzES0FFwup8c19s0hBeHZHXB/y5JMFegjvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "interim", "title": "Compute the slope of the perpendicular line:$${\\quad}m_{p}=-4$$", "steps": [ { "type": "step", "primary": "The perpendicular slope is the negative reciprocal of the given slope" }, { "type": "interim", "title": "$$\\frac{1}{4}m_{p}=-1{\\quad:\\quad}m_{p}=-4$$", "input": "\\frac{1}{4}m_{p}=-1", "steps": [ { "type": "interim", "title": "Multiply both sides by $$4$$", "input": "\\frac{1}{4}m_{p}=-1", "result": "m_{p}=-4", "steps": [ { "type": "step", "primary": "Multiply both sides by $$4$$", "result": "4\\cdot\\:\\frac{1}{4}m_{p}=4\\left(-1\\right)" }, { "type": "step", "primary": "Simplify", "result": "m_{p}=-4" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "m_{p}=-4" } ], "meta": { "interimType": "Line Equation Slope Perpendicular 0Eq" } }, { "type": "interim", "title": "Line with slope m=$$-4:{\\quad}y=-4x+b$$", "steps": [ { "type": "step", "primary": "Construct the line equation $$\\mathbf{y=mx+b}$$ with $$\\mathbf{m}=-4$$ and a general $$\\mathbf{b}$$", "result": "y=-4x+b" } ], "meta": { "interimType": "Line Equation General 1Eq" } }, { "type": "step", "result": "y=-4x+b" } ], "meta": { "solvingClass": "PreCalc" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{x}{4}+7", "displayFormula": "y=\\frac{x}{4}+7", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-4x+1", "displayFormula": "y=-4x+1", "attributes": { "color": "GRAY", "lineType": "NORMAL", "isAsymptote": false } } ] }, "functionChanges": [ { "origFormulaLatex": [ "\\frac{1}{4}x+7", "-4x+b" ], "finalFormulaLatex": [ "\\frac{x}{4}+7", "-4x+1" ], "plotTitle": "y=\\frac{1}{4}x+7, y=-4x+b", "paramsLatex": [ "b" ], "paramsReplacementsLatex": [ "1" ] } ], "localBoundingBox": { "xMin": -178.92, "xMax": 153.72, "yMin": -163.20499999999996, "yMax": 169.43499999999997 } }, "showViewLarger": true } }, "meta": { "showVerify": true } }