perpendicular 3x-4y=6

{ "query": { "display": "perpendicular $$3x-4y=6$$", "symbolab_question": "PRE_CALC#perpendicular 3x-4y=6" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Line Equations", "subTopic": "Perpendicular", "default": "y=-\\frac{4}{3}x+b", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "General line, perpendicular to $$3x-4y=6:{\\quad}y=-\\frac{4}{3}x+b$$", "steps": [ { "type": "interim", "title": "Find the slope of $$3x-4y=6:{\\quad}m=\\frac{3}{4}$$", "input": "3x-4y=6", "steps": [ { "type": "interim", "title": "Convert to slope intercept form:$${\\quad}y=\\frac{3}{4}x-\\frac{3}{2}$$", "input": "3x-4y=6", "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-4y=6", "result": "-4y=6-3x", "steps": [ { "type": "step", "primary": "Subtract $$3x$$ from both sides", "result": "3x-4y-3x=6-3x" }, { "type": "step", "primary": "Simplify", "result": "-4y=6-3x" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$-4$$", "input": "-4y=6-3x", "result": "y=\\frac{3}{4}x-\\frac{3}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$-4$$", "result": "\\frac{-4y}{-4}=\\frac{6}{-4}-\\frac{3x}{-4}" }, { "type": "step", "primary": "Simplify", "result": "y=\\frac{3}{4}x-\\frac{3}{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{3}{4}" } ], "meta": { "interimType": "Slope Equation Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMGyxztx1DIZ4Y9QoeLjXWS+oymo7oRHYXM600V1WY0Zh6pfF1z6umzUJTJvt+ojYZNjb/n1u0U6WX+tv49iLZKYG+xINzIApNgG0KtrDgdHE=" } }, { "type": "interim", "title": "Compute the slope of the perpendicular line:$${\\quad}m_{p}=-\\frac{4}{3}$$", "steps": [ { "type": "step", "primary": "The perpendicular slope is the negative reciprocal of the given slope" }, { "type": "interim", "title": "$$\\frac{3}{4}m_{p}=-1{\\quad:\\quad}m_{p}=-\\frac{4}{3}$$", "input": "\\frac{3}{4}m_{p}=-1", "steps": [ { "type": "interim", "title": "Multiply both sides by $$4$$", "input": "\\frac{3}{4}m_{p}=-1", "result": "3m_{p}=-4", "steps": [ { "type": "step", "primary": "Multiply both sides by $$4$$", "result": "4\\cdot\\:\\frac{3}{4}m_{p}=4\\left(-1\\right)" }, { "type": "step", "primary": "Simplify", "result": "3m_{p}=-4" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3m_{p}=-4", "result": "m_{p}=-\\frac{4}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3m_{p}}{3}=\\frac{-4}{3}" }, { "type": "step", "primary": "Simplify", "result": "m_{p}=-\\frac{4}{3}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "m_{p}=-\\frac{4}{3}" } ], "meta": { "interimType": "Line Equation Slope Perpendicular 0Eq" } }, { "type": "interim", "title": "Line with slope m=$$-\\frac{4}{3}:{\\quad}y=-\\frac{4}{3}x+b$$", "steps": [ { "type": "step", "primary": "Construct the line equation $$\\mathbf{y=mx+b}$$ with $$\\mathbf{m}=-\\frac{4}{3}$$ and a general $$\\mathbf{b}$$", "result": "y=-\\frac{4}{3}x+b" } ], "meta": { "interimType": "Line Equation General 1Eq" } }, { "type": "step", "result": "y=-\\frac{4}{3}x+b" } ], "meta": { "solvingClass": "PreCalc" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{3x}{4}-\\frac{3}{2}", "displayFormula": "y=\\frac{3x}{4}-\\frac{3}{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\frac{4x}{3}+1", "displayFormula": "y=-\\frac{4x}{3}+1", "attributes": { "color": "GRAY", "lineType": "NORMAL", "isAsymptote": false } } ] }, "functionChanges": [ { "origFormulaLatex": [ "\\frac{3}{4}x-\\frac{3}{2}", "-\\frac{4}{3}x+b" ], "finalFormulaLatex": [ "\\frac{3x}{4}-\\frac{3}{2}", "-\\frac{4x}{3}+1" ], "plotTitle": "y=\\frac{3}{4}x-\\frac{3}{2}, y=-\\frac{4}{3}x+b", "paramsLatex": [ "b" ], "paramsReplacementsLatex": [ "1" ] } ], "localBoundingBox": { "xMin": -10.979999999999999, "xMax": 12.78, "yMin": -12.5475, "yMax": 11.212499999999999 } }, "showViewLarger": true } }, "meta": { "showVerify": true } }