perpendicular 3x+6y=12

{ "query": { "display": "perpendicular $$3x+6y=12$$", "symbolab_question": "PRE_CALC#perpendicular 3x+6y=12" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Line Equations", "subTopic": "Perpendicular", "default": "y=2x+b", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "General line, perpendicular to $$3x+6y=12:{\\quad}y=2x+b$$", "steps": [ { "type": "interim", "title": "Find the slope of $$3x+6y=12:{\\quad}m=-\\frac{1}{2}$$", "input": "3x+6y=12", "steps": [ { "type": "interim", "title": "Convert to slope intercept form:$${\\quad}y=-\\frac{1}{2}x+2$$", "input": "3x+6y=12", "steps": [ { "type": "definition", "title": "Slope intercept form defininition", "text": "$$\\mathbf{y=mx+b}\\:$$is the slope intercept form of a line where $$\\mathbf{m}\\:$$is the slope and $$\\mathbf{b}\\:$$is the $$\\mathbf{y}\\:$$intercept" }, { "type": "interim", "title": "Move $$3x\\:$$to the right side", "input": "3x+6y=12", "result": "6y=12-3x", "steps": [ { "type": "step", "primary": "Subtract $$3x$$ from both sides", "result": "3x+6y-3x=12-3x" }, { "type": "step", "primary": "Simplify", "result": "6y=12-3x" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$6$$", "input": "6y=12-3x", "result": "y=-\\frac{1}{2}x+2", "steps": [ { "type": "step", "primary": "Divide both sides by $$6$$", "result": "\\frac{6y}{6}=\\frac{12}{6}-\\frac{3x}{6}" }, { "type": "step", "primary": "Simplify", "result": "y=-\\frac{1}{2}x+2" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "interimType": "Convert To Intercept Linear Equation 0Eq" } }, { "type": "step", "primary": "For a line equation for the form of $$\\mathbf{y=mx+b}$$, the slope is $$\\mathbf{m}$$", "result": "m=-\\frac{1}{2}" } ], "meta": { "interimType": "Slope Equation Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMGyxztx1DIZ4Y9QoeLjXWS7/bjAfIOXoilJZGjgn8vBbWwPs1+Gw97t4MeuaNjSYTOogQbTGFPHZ59tJQqmZojH0ZgKYn8iY0uPIy8tLxI8w=" } }, { "type": "interim", "title": "Compute the slope of the perpendicular line:$${\\quad}m_{p}=2$$", "steps": [ { "type": "step", "primary": "The perpendicular slope is the negative reciprocal of the given slope" }, { "type": "interim", "title": "$$\\left(-\\frac{1}{2}\\right)m_{p}=-1{\\quad:\\quad}m_{p}=2$$", "input": "\\left(-\\frac{1}{2}\\right)m_{p}=-1", "steps": [ { "type": "interim", "title": "Multiply both sides by $$-2$$", "input": "\\left(-\\frac{1}{2}\\right)m_{p}=-1", "result": "m_{p}=2", "steps": [ { "type": "step", "primary": "Multiply both sides by $$-2$$", "result": "\\left(-\\frac{1}{2}\\right)m_{p}\\left(-2\\right)=\\left(-1\\right)\\left(-2\\right)" }, { "type": "step", "primary": "Simplify", "result": "m_{p}=2" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7SZkIc84SBPqhfrySX6KICkxK4+R/h+2Nfzx9kofeI/9Zh3TrZjiMYl28xHyBITGVQEmfLuQ7ceONl+iO8D/69q2dgUdb/s3xRzA6qn/gnun6WHkl+QtvSKBY1ZLNzcZtHwKfDb3vWIaDKSwbm9Upm9j/RtTYJkwHRMpnvM8A6/UKXES3qYzjDMwmoojjBOzB31Gniku3XrbUM0ULk/V0R01wVxKhicy4GYIIbK9PkIdP2bFCKi/bOdAa8WQCR/QAhCHCKW/+tf9Tlqu392aOzaU5YxDLzW4dJ11H1VcrdkO0liQlMRTzJUmAIbvvxAK5bgJv9ltOmEssIBRWD1cpqd8dD0Lo6CIUNEYCqtsfAHgm/U1KHQsVoHy2nFpFSDpBjjVSpBZkaPfgpLmaKJuiP3x+pFb2rvhDUe2sIQFPphIdaAKcvHJaNe7Tjfb+jLNhaCj47GMC73PUEOQxvsSjQLd2HYQ40Rno5ziS/qB0Eqo=" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "m_{p}=2" } ], "meta": { "interimType": "Line Equation Slope Perpendicular 0Eq" } }, { "type": "interim", "title": "Line with slope m=$$2:{\\quad}y=2x+b$$", "steps": [ { "type": "step", "primary": "Construct the line equation $$\\mathbf{y=mx+b}$$ with $$\\mathbf{m}=2$$ and a general $$\\mathbf{b}$$", "result": "y=2x+b" } ], "meta": { "interimType": "Line Equation General 1Eq" } }, { "type": "step", "result": "y=2x+b" } ], "meta": { "solvingClass": "PreCalc" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=-\\frac{x}{2}+2", "displayFormula": "y=-\\frac{x}{2}+2", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=2x+1", "displayFormula": "y=2x+1", "attributes": { "color": "GRAY", "lineType": "NORMAL", "isAsymptote": false } } ] }, "functionChanges": [ { "origFormulaLatex": [ "-\\frac{1}{2}x+2", "2x+b" ], "finalFormulaLatex": [ "-\\frac{x}{2}+2", "2x+1" ], "plotTitle": "y=-\\frac{1}{2}x+2, y=2x+b", "paramsLatex": [ "b" ], "paramsReplacementsLatex": [ "1" ] } ], "localBoundingBox": { "xMin": -21.959999999999997, "xMax": 25.56, "yMin": -22.869999999999997, "yMax": 24.65 } }, "showViewLarger": true } }, "meta": { "showVerify": true } }