line (4,2),(-3,1)

{ "query": { "display": "line $$\\left(4,\\:2\\right),\\:\\left(-3,\\:1\\right)$$", "symbolab_question": "PRE_CALC#line (4,2),(-3,1)" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Line Equations", "subTopic": "Line", "default": "y=\\frac{1}{7}x+\\frac{10}{7}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Line passing through $$\\left(4,\\:2\\right)$$, $$\\left(-3,\\:1\\right):{\\quad}y=\\frac{1}{7}x+\\frac{10}{7}$$", "steps": [ { "type": "step", "primary": "Find the line $$\\mathbf{y=mx+b}$$ passing through $$\\left(4,\\:2\\right)$$, $$\\left(-3,\\:1\\right)$$" }, { "type": "interim", "title": "Compute the slope $$\\left(4,\\:2\\right),\\:\\left(-3,\\:1\\right):{\\quad}m=\\frac{1}{7}$$", "steps": [ { "type": "step", "primary": "Slope between two points: Slope$$=\\frac{y_2-y_1}{x_2-x_1}$$", "secondary": [ "$$\\left(x_1,\\:y_1\\right)=\\left(4,\\:2\\right),\\:\\left(x_2,\\:y_2\\right)=\\left(-3,\\:1\\right)$$" ], "result": "m=\\frac{1-2}{-3-4}" }, { "type": "step", "primary": "Refine", "result": "m=\\frac{1}{7}" } ], "meta": { "interimType": "Slope Two Points Top Level 4Eq" } }, { "type": "interim", "title": "Compute the $$y$$ intercept:$${\\quad}b=\\frac{10}{7}$$", "steps": [ { "type": "step", "primary": "Plug the slope $$\\frac{1}{7}$$ into $$y=mx+b$$", "result": "y=\\frac{1}{7}x+b" }, { "type": "step", "primary": "Plug in $$\\left(4,\\:2\\right)$$: $$\\quad\\:x=4,\\:y=2$$", "result": "2=\\frac{1}{7}\\cdot\\:4+b" }, { "type": "step", "primary": "Isolate $$b$$" }, { "type": "interim", "title": "$$2=\\frac{1}{7}\\cdot\\:4+b{\\quad:\\quad}b=\\frac{10}{7}$$", "input": "2=\\frac{1}{7}\\cdot\\:4+b", "steps": [ { "type": "step", "primary": "Switch sides", "result": "\\frac{1}{7}\\cdot\\:4+b=2" }, { "type": "interim", "title": "$$\\frac{1}{7}\\cdot\\:4=\\frac{4}{7}$$", "input": "\\frac{1}{7}\\cdot\\:4", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:4}{7}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:4=4$$", "result": "=\\frac{4}{7}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yGAHtoIDA1Tjh1HgZg+xL3p/BfzjvyNsWYj19qC8W+Crju+5Z51e/ZZSD3gRHwjB2w6Kfheb+lc+OoBmPSEC7mRLd2VwIqlBNByF6663syTQ4mLazZd24WQSmr6qmW36kPql8RlT7+zyODS9swnBPLCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "\\frac{4}{7}+b=2" }, { "type": "interim", "title": "Move $$\\frac{4}{7}\\:$$to the right side", "input": "\\frac{4}{7}+b=2", "result": "b=\\frac{10}{7}", "steps": [ { "type": "step", "primary": "Subtract $$\\frac{4}{7}$$ from both sides", "result": "\\frac{4}{7}+b-\\frac{4}{7}=2-\\frac{4}{7}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{4}{7}+b-\\frac{4}{7}=2-\\frac{4}{7}", "result": "b=\\frac{10}{7}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{4}{7}+b-\\frac{4}{7}:{\\quad}b$$", "input": "\\frac{4}{7}+b-\\frac{4}{7}", "steps": [ { "type": "step", "primary": "Add similar elements: $$\\frac{4}{7}-\\frac{4}{7}=0$$" }, { "type": "step", "result": "=b" } ], "meta": { "interimType": "Generic Simplify Specific 1Eq" } }, { "type": "interim", "title": "Simplify $$2-\\frac{4}{7}:{\\quad}\\frac{10}{7}$$", "input": "2-\\frac{4}{7}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$2=\\frac{2\\cdot\\:7}{7}$$", "result": "=\\frac{2\\cdot\\:7}{7}-\\frac{4}{7}" }, { "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\\:7-4}{7}" }, { "type": "interim", "title": "$$2\\cdot\\:7-4=10$$", "input": "2\\cdot\\:7-4", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:7=14$$", "result": "=14-4" }, { "type": "step", "primary": "Subtract the numbers: $$14-4=10$$", "result": "=10" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lbPF8zzq98jHtLgkSVtrS96GQqufR6tr2vPxOUv7H+8Khqe7+dajLA9h/DZSrCk/lEd8bz72O1EcTNHc6v8UnTB1Xqu/85fAvmEWTh0I6nw=" } }, { "type": "step", "result": "=\\frac{10}{7}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s75Jr7BP9+pdCeV7irHD64OACWKUbvV6WK3fDUgFtg3Q/ysS6ztOAKKFAEXmiY4WQmtMa+mfHJicgKoDyxJOYJOPC30sSftAIFS6Qkpy19Ikq76BZgUbAMdjd/rQeVX/4Yftyjy44UeUpwgJRweL56jg==" } }, { "type": "step", "result": "b=\\frac{10}{7}" } ], "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{10}{7}" } ], "meta": { "interimType": "Line Equation Find Intersection From Point 0Eq" } }, { "type": "step", "primary": "Construct the line equation $$\\mathbf{y=mx+b}$$ where $$\\mathbf{m}=\\frac{1}{7}$$ and $$\\mathbf{b}=\\frac{10}{7}$$", "result": "y=\\frac{1}{7}x+\\frac{10}{7}" } ], "meta": { "solvingClass": "PreCalc" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{x+10}{7}", "displayFormula": "y=\\frac{1}{7}x+\\frac{10}{7}", "derivativeFormula": "\\frac{1}{7}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(4,2)", "(-3,1)" ], "pointsDecimal": [ { "fst": 4, "snd": 2 }, { "fst": -3, "snd": 1 } ], "attributes": [ { "color": "BLACK", "labels": [ null ], "labelTypes": [ "SOLUTION" ], "labelColors": [ "BLACK" ] }, { "color": "BLACK", "labels": [ null ], "labelTypes": [ "SOLUTION" ], "labelColors": [ "BLACK" ] } ] }, "functionChanges": [ { "origFormulaLatex": [ "\\frac{1}{7}x+\\frac{10}{7}" ], "finalFormulaLatex": [ "\\frac{x+10}{7}" ], "plotTitle": "y=\\frac{1}{7}x+\\frac{10}{7}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -38.16, "xMax": 29.159999999999997, "yMin": -2.7894285714285716, "yMax": 4.060857142857143 } }, "showViewLarger": true } }, "meta": { "showVerify": true } }