line (3, 1/4),(3/2 ,7)

{ "query": { "display": "line $$\\left(3,\\:\\frac{1}{4}\\right),\\:\\left(\\frac{3}{2},\\:7\\right)$$", "symbolab_question": "PRE_CALC#line (3,\\frac{1}{4}),(\\frac{3}{2},7)" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Line Equations", "subTopic": "Line", "default": "y=-\\frac{9}{2}x+\\frac{55}{4}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Line passing through $$\\left(3,\\:\\frac{1}{4}\\right)$$, $$\\left(\\frac{3}{2},\\:7\\right):{\\quad}y=-\\frac{9}{2}x+\\frac{55}{4}$$", "steps": [ { "type": "step", "primary": "Find the line $$\\mathbf{y=mx+b}$$ passing through $$\\left(3,\\:\\frac{1}{4}\\right)$$, $$\\left(\\frac{3}{2},\\:7\\right)$$" }, { "type": "interim", "title": "Compute the slope $$\\left(3,\\:\\frac{1}{4}\\right),\\:\\left(\\frac{3}{2},\\:7\\right):{\\quad}m=-\\frac{9}{2}$$", "steps": [ { "type": "step", "primary": "Slope between two points: Slope$$=\\frac{y_2-y_1}{x_2-x_1}$$", "secondary": [ "$$\\left(x_1,\\:y_1\\right)=\\left(3,\\:\\frac{1}{4}\\right),\\:\\left(x_2,\\:y_2\\right)=\\left(\\frac{3}{2},\\:7\\right)$$" ], "result": "m=\\frac{7-\\frac{1}{4}}{\\frac{3}{2}-3}" }, { "type": "step", "primary": "Refine", "result": "m=-\\frac{9}{2}" } ], "meta": { "interimType": "Slope Two Points Top Level 4Eq" } }, { "type": "interim", "title": "Compute the $$y$$ intercept:$${\\quad}b=\\frac{55}{4}$$", "steps": [ { "type": "step", "primary": "Plug the slope $$-\\frac{9}{2}$$ into $$y=mx+b$$", "result": "y=\\left(-\\frac{9}{2}\\right)x+b" }, { "type": "step", "primary": "Plug in $$\\left(3,\\:\\frac{1}{4}\\right)$$: $$\\quad\\:x=3,\\:y=\\frac{1}{4}$$", "result": "\\frac{1}{4}=\\left(-\\frac{9}{2}\\right)\\cdot\\:3+b" }, { "type": "step", "primary": "Isolate $$b$$" }, { "type": "interim", "title": "$$\\frac{1}{4}=\\left(-\\frac{9}{2}\\right)\\cdot\\:3+b{\\quad:\\quad}b=\\frac{55}{4}$$", "input": "\\frac{1}{4}=\\left(-\\frac{9}{2}\\right)\\cdot\\:3+b", "steps": [ { "type": "step", "primary": "Switch sides", "result": "\\left(-\\frac{9}{2}\\right)\\cdot\\:3+b=\\frac{1}{4}" }, { "type": "interim", "title": "Move $$\\left(-\\frac{9}{2}\\right)3\\:$$to the right side", "input": "\\left(-\\frac{9}{2}\\right)\\cdot\\:3+b=\\frac{1}{4}", "result": "b=\\frac{55}{4}", "steps": [ { "type": "step", "primary": "Subtract $$\\left(-\\frac{9}{2}\\right)3$$ from both sides", "result": "\\left(-\\frac{9}{2}\\right)\\cdot\\:3+b-\\left(-\\frac{9}{2}\\right)\\cdot\\:3=\\frac{1}{4}-\\left(-\\frac{9}{2}\\right)\\cdot\\:3" }, { "type": "interim", "title": "Simplify", "input": "\\left(-\\frac{9}{2}\\right)\\cdot\\:3+b-\\left(-\\frac{9}{2}\\right)\\cdot\\:3=\\frac{1}{4}-\\left(-\\frac{9}{2}\\right)\\cdot\\:3", "result": "b=\\frac{55}{4}", "steps": [ { "type": "interim", "title": "Simplify $$\\left(-\\frac{9}{2}\\right)\\cdot\\:3+b-\\left(-\\frac{9}{2}\\right)\\cdot\\:3:{\\quad}b$$", "input": "\\left(-\\frac{9}{2}\\right)\\cdot\\:3+b-\\left(-\\frac{9}{2}\\right)\\cdot\\:3", "steps": [ { "type": "step", "primary": "Add similar elements: $$\\left(-\\frac{9}{2}\\right)3-\\left(-\\frac{9}{2}\\right)3=0$$" }, { "type": "step", "result": "=b" } ], "meta": { "interimType": "Generic Simplify Specific 1Eq" } }, { "type": "interim", "title": "Simplify $$\\frac{1}{4}-\\left(-\\frac{9}{2}\\right)\\cdot\\:3:{\\quad}\\frac{55}{4}$$", "input": "\\frac{1}{4}-\\left(-\\frac{9}{2}\\right)\\cdot\\:3", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{1}{4}+\\frac{9}{2}\\cdot\\:3" }, { "type": "interim", "title": "$$\\frac{9}{2}\\cdot\\:3=\\frac{27}{2}$$", "input": "\\frac{9}{2}\\cdot\\:3", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{9\\cdot\\:3}{2}" }, { "type": "step", "primary": "Multiply the numbers: $$9\\cdot\\:3=27$$", "result": "=\\frac{27}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s75f89wMyJWjLGHcFMeIJPeiPpFLppSOOOwknMsNOrv36rju+5Z51e/ZZSD3gRHwjBmdVdt6v3kVQvG4NTWH+a7D/L0MoYg+CUn6oyL3EO7Yp4qzHyVKJTIKQJD/FNV9+KYuDW5QiJ3tlJN1Gkp3CcJY8BPOx0wlsgFN8qUa6AzA0=" } }, { "type": "step", "result": "=\\frac{1}{4}+\\frac{27}{2}" }, { "type": "interim", "title": "Least Common Multiplier of $$4,\\:2:{\\quad}4$$", "input": "4,\\:2", "steps": [ { "type": "definition", "title": "Least Common Multiplier (LCM)", "text": "The LCM of $$a,\\:b$$ is the smallest positive number that is divisible by both $$a$$ and $$b$$" }, { "type": "interim", "title": "Prime factorization of $$4:{\\quad}2\\cdot\\:2$$", "input": "4", "steps": [ { "type": "step", "primary": "$$4\\:$$divides by $$2\\quad\\:4=2\\cdot\\:2$$", "result": "=2\\cdot\\:2" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRsG/uC0ndYtZpJL4uAxK7FI/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp39fF/zAtU5baHQ1hwgXA+n" } }, { "type": "interim", "title": "Prime factorization of $$2:{\\quad}2$$", "input": "2", "steps": [ { "type": "step", "primary": "$$2$$ is a prime number, therefore no factorization is possible", "result": "=2" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRl8ZboA8wPLg0yhI4RzfjFw/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp1+G9v2aKasChgV65VW8cTW" } }, { "type": "step", "primary": "Multiply each factor the greatest number of times it occurs in either $$4$$ or $$2$$", "result": "=2\\cdot\\:2" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=4" } ], "meta": { "solvingClass": "LCM", "interimType": "LCM Top 1Eq" } }, { "type": "interim", "title": "Adjust Fractions based on the LCM", "steps": [ { "type": "step", "primary": "Multiply each numerator by the same amount needed to multiply its<br/>corresponding denominator to turn it into the LCM $$4$$" }, { "type": "step", "primary": "For $$\\frac{27}{2}:\\:$$multiply the denominator and numerator by $$2$$", "result": "\\frac{27}{2}=\\frac{27\\cdot\\:2}{2\\cdot\\:2}=\\frac{54}{4}" } ], "meta": { "interimType": "LCD Adjust Fractions 1Eq" } }, { "type": "step", "result": "=\\frac{1}{4}+\\frac{54}{4}" }, { "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{1+54}{4}" }, { "type": "step", "primary": "Add the numbers: $$1+54=55$$", "result": "=\\frac{55}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7fxEGHi9ShAHm1TGGbqmH4i3I2Na2mxVh4gFUWcr8Imkj6RS6aUjjjsJJzLDTq79+q47vuWedXv2WUg94ER8IwRQ1JbmSWsoOjV5dLPo/1Fk/y9DKGIPglJ+qMi9xDu2KaRI7GCp0HQz+zDw23axddHqnu5blpT1gqyZBFSXBFqsm2T4Iivi+LHdpH0v5a67q6sRpHVg8NBGJ+/B6+Sq5+g==" } }, { "type": "step", "result": "b=\\frac{55}{4}" } ], "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{55}{4}" } ], "meta": { "interimType": "Line Equation Find Intersection From Point 0Eq" } }, { "type": "step", "primary": "Construct the line equation $$\\mathbf{y=mx+b}$$ where $$\\mathbf{m}=-\\frac{9}{2}$$ and $$\\mathbf{b}=\\frac{55}{4}$$", "result": "y=-\\frac{9}{2}x+\\frac{55}{4}" } ], "meta": { "solvingClass": "PreCalc" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=-\\frac{9x}{2}+\\frac{55}{4}", "displayFormula": "y=-\\frac{9}{2}x+\\frac{55}{4}", "derivativeFormula": "-\\frac{9}{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(3,\\frac{1}{4})", "(\\frac{3}{2},7)" ], "pointsDecimal": [ { "fst": 3, "snd": 0.25 }, { "fst": 1.5, "snd": 7 } ], "attributes": [ { "color": "BLACK", "labels": [ null ], "labelTypes": [ "SOLUTION" ], "labelColors": [ "BLACK" ] }, { "color": "BLACK", "labels": [ null ], "labelTypes": [ "SOLUTION" ], "labelColors": [ "BLACK" ] } ] }, "functionChanges": [ { "origFormulaLatex": [ "-\\frac{9}{2}x+\\frac{55}{4}" ], "finalFormulaLatex": [ "-\\frac{9x}{2}+\\frac{55}{4}" ], "plotTitle": 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