parallel 2x-3y=9,(2,2)

{ "query": { "display": "parallel $$2x-3y=9,\\:\\left(2,\\:2\\right)$$", "symbolab_question": "PRE_CALC#parallel 2x-3y=9,(2,2)" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Line Equations", "subTopic": "Parallel", "default": "y=\\frac{2}{3}x+\\frac{2}{3}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Parallel to $$2x-3y=9$$, and passing through $$\\left(2,\\:2\\right):{\\quad}y=\\frac{2}{3}x+\\frac{2}{3}$$", "steps": [ { "type": "step", "primary": "Find the line $$\\mathbf{y=mx+b}$$ parallel to $$2x-3y=9$$ that passes through $$\\left(2,\\:2\\right)$$" }, { "type": "interim", "title": "Find the slope of $$2x-3y=9:{\\quad}m=\\frac{2}{3}$$", "input": "2x-3y=9", "steps": [ { "type": "interim", "title": "Convert to slope intercept form:$${\\quad}y=\\frac{2}{3}x-3$$", "input": "2x-3y=9", "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 $$2x\\:$$to the right side", "input": "2x-3y=9", "result": "-3y=9-2x", "steps": [ { "type": "step", "primary": "Subtract $$2x$$ from both sides", "result": "2x-3y-2x=9-2x" }, { "type": "step", "primary": "Simplify", "result": "-3y=9-2x" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$-3$$", "input": "-3y=9-2x", "result": "y=\\frac{2}{3}x-3", "steps": [ { "type": "step", "primary": "Divide both sides by $$-3$$", "result": "\\frac{-3y}{-3}=\\frac{9}{-3}-\\frac{2x}{-3}" }, { "type": "step", "primary": "Simplify", "result": "y=\\frac{2}{3}x-3" } ], "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{2}{3}" } ], "meta": { "interimType": "Slope Equation Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMGyxztx1DIZ4Y9QoeLjXWS+QbBLKkQl9cs/+Tw1OVN196pfF1z6umzUJTJvt+ojYZNjb/n1u0U6WX+tv49iLZKeSnwXBGILRm1dhmm/q+32s=" } }, { "type": "interim", "title": "Compute the $$y$$ intercept:$${\\quad}b=\\frac{2}{3}$$", "steps": [ { "type": "step", "primary": "The slopes of parallel lines are equal" }, { "type": "step", "primary": "Plug the slope $$\\frac{2}{3}$$ into $$y=mx+b$$", "result": "y=\\frac{2}{3}x+b" }, { "type": "step", "primary": "Plug in $$\\left(2,\\:2\\right)$$: $$\\quad\\:x=2,\\:y=2$$", "result": "2=\\frac{2}{3}\\cdot\\:2+b" }, { "type": "step", "primary": "Isolate $$b$$" }, { "type": "interim", "title": "$$2=\\frac{2}{3}\\cdot\\:2+b{\\quad:\\quad}b=\\frac{2}{3}$$", "input": "2=\\frac{2}{3}\\cdot\\:2+b", "steps": [ { "type": "step", "primary": "Switch sides", "result": "\\frac{2}{3}\\cdot\\:2+b=2" }, { "type": "interim", "title": "$$\\frac{2}{3}\\cdot\\:2=\\frac{4}{3}$$", "input": "\\frac{2}{3}\\cdot\\:2", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{2\\cdot\\:2}{3}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{4}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7dqk73cbRXRvyGMK3+H/qoe30Fwl9QKPJxyO/TFRCb5Grju+5Z51e/ZZSD3gRHwjBjWsIGWMvuWc1k1QEbWgxuWRLd2VwIqlBNByF6663sySbSig4yd/ciJMw5zULPejyHyUBRrmvP3KbrWkOg8VcLrCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "\\frac{4}{3}+b=2" }, { "type": "interim", "title": "Move $$\\frac{4}{3}\\:$$to the right side", "input": "\\frac{4}{3}+b=2", "result": "b=\\frac{2}{3}", "steps": [ { "type": "step", "primary": "Subtract $$\\frac{4}{3}$$ from both sides", "result": "\\frac{4}{3}+b-\\frac{4}{3}=2-\\frac{4}{3}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{4}{3}+b-\\frac{4}{3}=2-\\frac{4}{3}", "result": "b=\\frac{2}{3}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{4}{3}+b-\\frac{4}{3}:{\\quad}b$$", "input": "\\frac{4}{3}+b-\\frac{4}{3}", "steps": [ { "type": "step", "primary": "Add similar elements: $$\\frac{4}{3}-\\frac{4}{3}=0$$" }, { "type": "step", "result": "=b" } ], "meta": { "interimType": "Generic Simplify Specific 1Eq" } }, { "type": "interim", "title": "Simplify $$2-\\frac{4}{3}:{\\quad}\\frac{2}{3}$$", "input": "2-\\frac{4}{3}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$2=\\frac{2\\cdot\\:3}{3}$$", "result": "=\\frac{2\\cdot\\:3}{3}-\\frac{4}{3}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{2\\cdot\\:3-4}{3}" }, { "type": "interim", "title": "$$2\\cdot\\:3-4=2$$", "input": "2\\cdot\\:3-4", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=6-4" }, { "type": "step", "primary": "Subtract the numbers: $$6-4=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ucNlOgK1FT1BGMFxbzENYN6GQqufR6tr2vPxOUv7H+9hLZhL3VORCH2mYsYpGjlN1SvY/eJGzvEmlW7hoPETDmUYTdxv+A+H+X6bcINIo94=" } }, { "type": "step", "result": "=\\frac{2}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XM0eP1aYeYpWKObQrptUyQCWKUbvV6WK3fDUgFtg3Q/ysS6ztOAKKFAEXmiY4WQmx14CeeoVSZQ/AQtcrmsIf9bA+zX4bD3u3gx65o2NJhP1CVdov1jXsz8NpLG+sfRzhE2ewgPRYiAGUYOUr9tERw==" } }, { "type": "step", "result": "b=\\frac{2}{3}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "b=\\frac{2}{3}" } ], "meta": { "interimType": "Line Equation Find Intersection From Point 0Eq" } }, { "type": "step", "primary": "Construct the line equation $$\\mathbf{y=mx+b}$$ where $$\\mathbf{m}=\\frac{2}{3}$$ and $$\\mathbf{b}=\\frac{2}{3}$$", "result": "y=\\frac{2}{3}x+\\frac{2}{3}" } ], "meta": { "solvingClass": "PreCalc" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{2x}{3}-3", "displayFormula": "y=\\frac{2x}{3}-3", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=\\frac{2x+2}{3}", "displayFormula": "y=\\frac{2x+2}{3}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(2,2)" ], "pointsDecimal": [ { "fst": 2, "snd": 2 } ], "attributes": [ { "color": "PURPLE", "labels": [ null ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] } ] }, "functionChanges": [ { "origFormulaLatex": [ "\\frac{2}{3}x-3", "\\frac{2}{3}x+\\frac{2}{3}" ], "finalFormulaLatex": [ "\\frac{2x}{3}-3", "\\frac{2x+2}{3}" ], "plotTitle": "y=\\frac{2}{3}x-3, y=\\frac{2}{3}x+\\frac{2}{3}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -24.705, "xMax": 28.755000000000003, "yMin": -8.527800000000001, "yMax": 5.857800000000001 } }, "showViewLarger": true } }, "meta": { "showVerify": true } }