five years ago i was 5 times older than my son. in 8 years time i will be 3 times older than my son. how old am i today

{ "query": { "display": "five years ago i was 5 times older than my son. in 8 years time i will be 3 times older than my son. how old am i today", "symbolab_question": "#five years ago i was 5 times older than my son. in 8 years time i will be 3 times older than my son. how old am i today" }, "solution": { "level": "PERFORMED", "subject": "Word Problems", "topic": "Age", "subTopic": "Other", "default": "\\mathrm{my}\\mathrm{\\:age\\:is:}\\:70" }, "steps": { "type": "interim", "title": "5 years ago i was 5 times older than my son.<br/>in 8 years time i will be 3 times older than my son.<br/>how old am i today<br/> <br/>my age is: $$70$$", "steps": [ { "type": "interim", "title": "Translate the problem into an equation:$${\\quad}x-5=5\\left(y-5\\right),\\:x+8=3\\left(y+8\\right)$$", "steps": [ { "type": "step", "primary": "Represent ages in terms of $$x\\land\\:y:$$", "result": "\\mathrm{me}\\::\\:x<br/>\\mathrm{Son}\\::\\:y" }, { "type": "step", "primary": "Ages 5 years ago", "result": "\\mathrm{me}\\::\\:x-5<br/>\\mathrm{Son}\\::\\:y-5" }, { "type": "step", "primary": "Ages in 8 years", "result": "\\mathrm{me}\\::\\:x+8<br/>\\mathrm{Son}\\::\\:y+8" }, { "type": "step", "primary": "Write as system of equations", "result": "x-5=5\\left(y-5\\right)<br/>x+8=3\\left(y+8\\right)" } ] }, { "type": "interim", "title": "$$x-5=5\\left(y-5\\right),\\:x+8=3\\left(y+8\\right){\\quad:\\quad}x=70,\\:y=18$$", "steps": [ { "type": "step", "result": "\\begin{bmatrix}x-5=5\\left(y-5\\right)\\\\x+8=3\\left(y+8\\right)\\end{bmatrix}" }, { "type": "interim", "title": "Isolate $$x\\:$$for $$x-5=5\\cdot\\:\\left(y-5\\right):{\\quad}x=5y-20$$", "input": "x-5=5\\left(y-5\\right)", "steps": [ { "type": "interim", "title": "Move $$5\\:$$to the right side", "input": "x-5=5\\left(y-5\\right)", "result": "x=5y-20", "steps": [ { "type": "step", "primary": "Add $$5$$ to both sides", "result": "x-5+5=5\\left(y-5\\right)+5" }, { "type": "step", "primary": "Simplify", "result": "x=5y-20" } ], "meta": { "gptData": "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" } } ], "meta": { "solvingClass": "Equations" } }, { "type": "step", "primary": "Substitute $$x=5y-20$$", "result": "\\begin{bmatrix}5y-20+8=3\\left(y+8\\right)\\end{bmatrix}" }, { "type": "interim", "title": "Simplify", "input": "5y-20+8=3\\left(y+8\\right)", "steps": [ { "type": "interim", "title": "Simplify $$5y-20+8:{\\quad}5y-12$$", "input": "5y-20+8", "steps": [ { "type": "step", "primary": "Add/Subtract the numbers: $$-20+8=-12$$", "result": "=5y-12" } ], "meta": { "solvingClass": "Solver" } }, { "type": "step", "result": "5y-12=3\\left(y+8\\right)" } ], "meta": { "solvingClass": "Solver" } }, { "type": "step", "result": "\\begin{bmatrix}5y-12=3\\left(y+8\\right)\\end{bmatrix}" }, { "type": "interim", "title": "Isolate $$y\\:$$for $$5y-12=3\\left(y+8\\right):{\\quad}y=18$$", "input": "5y-12=3\\left(y+8\\right)", "steps": [ { "type": "interim", "title": "Expand $$3\\left(y+8\\right):{\\quad}3y+24$$", "input": "3\\left(y+8\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=3,\\:b=y,\\:c=8$$" ], "result": "=3y+3\\cdot\\:8", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:8=24$$", "result": "=3y+24" } ], "meta": { "solvingClass": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vGDR3lLS2TqSYZmoGJkLDgsBmUOGgB0VAXoAXik9qC9EoApuujwRq/Ms3wa4gNT89ZVSO0B13oSpK+zJfqv7kB7MWKURN+43KCOzRc+RXaE086DSB1b1NEjxdwhNTxqT" } }, { "type": "step", "result": "5y-12=3y+24" }, { "type": "interim", "title": "Move $$12\\:$$to the right side", "input": "5y-12=3y+24", "result": "5y=3y+36", "steps": [ { "type": "step", "primary": "Add $$12$$ to both sides", "result": "5y-12+12=3y+24+12" }, { "type": "step", "primary": "Simplify", "result": "5y=3y+36" } ], "meta": { "gptData": "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" } }, { "type": "interim", "title": "Move $$3y\\:$$to the left side", "input": "5y=3y+36", "result": "2y=36", "steps": [ { "type": "step", "primary": "Subtract $$3y$$ from both sides", "result": "5y-3y=3y+36-3y" }, { "type": "step", "primary": "Simplify", "result": "2y=36" } ], "meta": { "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yhU883qBpVI8EPRByBJmqUNjEtO5EuzK5/hrJfagkSp04xZIV4JgAfKcH2uXMbGc88UxTzlGSTEJ2AYfegcR+3w+vD12EwiukyDhrtX/OqpbwVz5T7/Dyunz8G+H2e0kUsD+pIkOCIZyIU4q8atffPpdu80I8/0M2UIpmdiK+fzpSJIDD6j6cnbjyilKFfQEhnCTXiYdi8kfyoSqnBXcwV3yIeazVTHV22QGvC0c9M3MkJoKuab64EQlbP+PDl2DHg16QmZuySiZ7alsCkoKM23log7j8sDEg4mKhnyQg7in6WU7ybUlSwezFhLKvkKNVvuqUoYkKmBK6fSR2cNlgkfqfT3HSxldmNt6rUcfvySCyrAO9zMQDZyjEiWM7qI5HNLamESyiXkEVNU4Be3yXlX3rP7J381OCLdf7C7WWRDvbBmbuQNTF0TphKZ8Ruvaw5GGcrtZ5kdS9au3iLahhkD8cdBg31rv1aJ4Pcpkmg+fMYgc/JaxRvxnRTG/TM3C" } }, { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2y=36", "result": "y=18", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2y}{2}=\\frac{36}{2}" }, { "type": "step", "primary": "Simplify", "result": "y=18" } ], "meta": { "gptData": "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" } } ], "meta": { "solvingClass": "Equations" } }, { "type": "step", "primary": "For $$x=5y-20$$", "secondary": [ "Substitute $$y=18$$" ], "result": "x=5\\cdot\\:18-20" }, { "type": "step", "primary": "Simplify", "result": "x=70" }, { "type": "step", "primary": "The solutions to the system of equations are:", "result": "x=70,\\:y=18" } ], "meta": { "solvingClass": "System of Equations" } }, { "type": "step", "result": "\\mathrm{my}\\mathrm{\\:age\\:is:}\\:70" } ] } }