perpendicular y=5x+2,\at x=1

{ "query": { "display": "perpendicular $$y=5x+2,\\:\\at\\:x=1$$", "symbolab_question": "PRE_CALC#perpendicular y=5x+2,\\at x=1" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Line Equations", "subTopic": "Perpendicular", "default": "y=-\\frac{1}{5}x+\\frac{36}{5}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Perpendicular to $$y=5x+2$$, and passing through $$x=1:{\\quad}y=-\\frac{1}{5}x+\\frac{36}{5}$$", "steps": [ { "type": "step", "primary": "Find the line $$\\mathbf{y=mx+b}$$ perpendicular to $$y=5x+2$$ that passes through $$x=1$$" }, { "type": "interim", "title": "Find the perpendicular point:$${\\quad}\\left(1,\\:7\\right)$$", "steps": [ { "type": "step", "primary": "Plug $$x=1$$ into the equation $$y=5x+2$$", "result": "y=5\\cdot\\:1+2" }, { "type": "step", "primary": "Solve $$y$$", "result": "y=7" } ], "meta": { "interimType": "Perpendicular Find Point 0Eq" } }, { "type": "interim", "title": "Find the slope of $$y=5x+2:{\\quad}m=5$$", "input": "y=5x+2", "steps": [ { "type": "step", "primary": "For a line equation for the form of $$\\mathbf{y=mx+b}$$, the slope is $$\\mathbf{m}$$", "result": "m=5" } ], "meta": { "interimType": "Slope Equation Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMGyxztx1DIZ4Y9QoeLjXWS8GT/xuohWmSjjrZH3SomS5N5Aod6Hr1Lp2e/29KhSgU2EQBSOB7bD5kYsyTsTrqcUU40jHTIbN0bQwqC5UymbY=" } }, { "type": "interim", "title": "Compute the slope of the perpendicular line:$${\\quad}m_{p}=-\\frac{1}{5}$$", "steps": [ { "type": "step", "primary": "The perpendicular slope is the negative reciprocal of the given slope" }, { "type": "interim", "title": "$$5m_{p}=-1{\\quad:\\quad}m_{p}=-\\frac{1}{5}$$", "input": "5m_{p}=-1", "steps": [ { "type": "interim", "title": "Divide both sides by $$5$$", "input": "5m_{p}=-1", "result": "m_{p}=-\\frac{1}{5}", "steps": [ { "type": "step", "primary": "Divide both sides by $$5$$", "result": "\\frac{5m_{p}}{5}=\\frac{-1}{5}" }, { "type": "step", "primary": "Simplify", "result": "m_{p}=-\\frac{1}{5}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "m_{p}=-\\frac{1}{5}" } ], "meta": { "interimType": "Line Equation Slope Perpendicular 0Eq" } }, { "type": "interim", "title": "Find the line with slope m=$$-\\frac{1}{5}$$ and passing through $$\\left(1,\\:7\\right):{\\quad}y=-\\frac{1}{5}x+\\frac{36}{5}$$", "steps": [ { "type": "step", "primary": "Compute the line equation $$\\mathbf{y=mx+b}$$ for slope m=$$-\\frac{1}{5}$$ and passing through $$\\left(1,\\:7\\right)$$" }, { "type": "interim", "title": "Compute the $$y$$ intercept:$${\\quad}b=\\frac{36}{5}$$", "steps": [ { "type": "step", "primary": "Plug the slope $$-\\frac{1}{5}$$ into $$y=mx+b$$", "result": "y=\\left(-\\frac{1}{5}\\right)x+b" }, { "type": "step", "primary": "Plug in $$\\left(1,\\:7\\right)$$: $$\\quad\\:x=1,\\:y=7$$", "result": "7=\\left(-\\frac{1}{5}\\right)\\cdot\\:1+b" }, { "type": "step", "primary": "Isolate $$b$$" }, { "type": "interim", "title": "$$7=\\left(-\\frac{1}{5}\\right)\\cdot\\:1+b{\\quad:\\quad}b=\\frac{36}{5}$$", "input": "7=\\left(-\\frac{1}{5}\\right)\\cdot\\:1+b", "steps": [ { "type": "step", "primary": "Switch sides", "result": "\\left(-\\frac{1}{5}\\right)\\cdot\\:1+b=7" }, { "type": "interim", "title": "Move $$\\left(-\\frac{1}{5}\\right)1\\:$$to the right side", "input": "\\left(-\\frac{1}{5}\\right)\\cdot\\:1+b=7", "result": "b=\\frac{36}{5}", "steps": [ { "type": "step", "primary": "Subtract $$\\left(-\\frac{1}{5}\\right)1$$ from both sides", "result": "\\left(-\\frac{1}{5}\\right)\\cdot\\:1+b-\\left(-\\frac{1}{5}\\right)\\cdot\\:1=7-\\left(-\\frac{1}{5}\\right)\\cdot\\:1" }, { "type": "interim", "title": "Simplify", "input": "\\left(-\\frac{1}{5}\\right)\\cdot\\:1+b-\\left(-\\frac{1}{5}\\right)\\cdot\\:1=7-\\left(-\\frac{1}{5}\\right)\\cdot\\:1", "result": "b=\\frac{36}{5}", "steps": [ { "type": "interim", "title": "Simplify $$\\left(-\\frac{1}{5}\\right)\\cdot\\:1+b-\\left(-\\frac{1}{5}\\right)\\cdot\\:1:{\\quad}b$$", "input": "\\left(-\\frac{1}{5}\\right)\\cdot\\:1+b-\\left(-\\frac{1}{5}\\right)\\cdot\\:1", "steps": [ { "type": "step", "primary": "Add similar elements: $$\\left(-\\frac{1}{5}\\right)1-\\left(-\\frac{1}{5}\\right)1=0$$" }, { "type": "step", "result": "=b" } ], "meta": { "interimType": "Generic Simplify Specific 1Eq" } }, { "type": "interim", "title": "Simplify $$7-\\left(-\\frac{1}{5}\\right)\\cdot\\:1:{\\quad}\\frac{36}{5}$$", "input": "7-\\left(-\\frac{1}{5}\\right)\\cdot\\:1", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=7+\\frac{1}{5}\\cdot\\:1" }, { "type": "step", "primary": "Multiply: $$\\frac{1}{5}\\cdot\\:1=\\frac{1}{5}$$", "result": "=7+\\frac{1}{5}" }, { "type": "step", "primary": "Convert element to fraction: $$7=\\frac{7\\cdot\\:5}{5}$$", "result": "=\\frac{7\\cdot\\:5}{5}+\\frac{1}{5}" }, { "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{7\\cdot\\:5+1}{5}" }, { "type": "interim", "title": "$$7\\cdot\\:5+1=36$$", "input": "7\\cdot\\:5+1", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$7\\cdot\\:5=35$$", "result": "=35+1" }, { "type": "step", "primary": "Add the numbers: $$35+1=36$$", "result": "=36" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AkdCiUHyVS9txmwC03XPU96GQqufR6tr2vPxOUv7H++OXB82dHM+bqH6dNBydiO+jOd1CXNtCNcU0eHbLrylVpT7EkkDd14uMXvYLC7YaQI=" } }, { "type": "step", "result": "=\\frac{36}{5}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YsaySRl0jEtQTtrr+ROgJiKvK5r5toWrLucw3VRy9Uzdd47a0hQ8flDbGsI5To1drfMRPynLSfP3VYe1OQldRC9CRMlutmgYJ2I1aTzCgWKLGmNnLPWGf9PH3lpmjoJIQER8p7VO8TGXPxNxNpjJ9l9LLeX3TggbK3rH5Sa5zaozXPlZOPbrJTojBoCz7go7" } }, { "type": "step", "result": "b=\\frac{36}{5}" } ], "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{36}{5}" } ], "meta": { "interimType": "Line Equation Find Intersection From Point 0Eq" } }, { "type": "step", "primary": "Construct the line equation $$\\mathbf{y=mx+b}$$ where $$\\mathbf{m}=-\\frac{1}{5}$$ and $$\\mathbf{b}=\\frac{36}{5}$$", "result": "y=-\\frac{1}{5}x+\\frac{36}{5}" } ], "meta": { "interimType": "Line Equation Slope Point 6Eq" } }, { "type": "step", "result": "y=-\\frac{1}{5}x+\\frac{36}{5}" } ], "meta": { "solvingClass": "PreCalc" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=5x+2", "displayFormula": "y=5x+2", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=\\frac{-x+36}{5}", "displayFormula": "y=\\frac{-x+36}{5}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(1,7)" ], "pointsDecimal": [ { "fst": 1, "snd": 7 } ], "attributes": [ { "color": "PURPLE", "labels": [ null ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] } ] }, "functionChanges": [ { "origFormulaLatex": [ 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