3/4+7/10 y= 7/20-2/5 y

{ "query": { "display": "$$\\frac{3}{4}+\\frac{7}{10}y=\\frac{7}{20}-\\frac{2}{5}y$$", "symbolab_question": "EQUATION#\\frac{3}{4}+\\frac{7}{10}y=\\frac{7}{20}-\\frac{2}{5}y" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Equations", "subTopic": "Linear", "default": "y=-\\frac{4}{11}", "decimal": "y=-0.36363…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{3}{4}+\\frac{7}{10}y=\\frac{7}{20}-\\frac{2}{5}y{\\quad:\\quad}y=-\\frac{4}{11}$$", "input": "\\frac{3}{4}+\\frac{7}{10}y=\\frac{7}{20}-\\frac{2}{5}y", "steps": [ { "type": "interim", "title": "Move $$\\frac{3}{4}\\:$$to the right side", "input": "\\frac{3}{4}+\\frac{7}{10}y=\\frac{7}{20}-\\frac{2}{5}y", "result": "\\frac{7}{10}y=-\\frac{2\\left(y+1\\right)}{5}", "steps": [ { "type": "step", "primary": "Subtract $$\\frac{3}{4}$$ from both sides", "result": "\\frac{3}{4}+\\frac{7}{10}y-\\frac{3}{4}=\\frac{7}{20}-\\frac{2}{5}y-\\frac{3}{4}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{3}{4}+\\frac{7}{10}y-\\frac{3}{4}=\\frac{7}{20}-\\frac{2}{5}y-\\frac{3}{4}", "result": "\\frac{7}{10}y=-\\frac{2\\left(y+1\\right)}{5}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3}{4}+\\frac{7}{10}y-\\frac{3}{4}:{\\quad}\\frac{7}{10}y$$", "input": "\\frac{3}{4}+\\frac{7}{10}y-\\frac{3}{4}", "steps": [ { "type": "step", "primary": "Add similar elements: $$\\frac{3}{4}-\\frac{3}{4}=0$$" }, { "type": "step", "result": "=\\frac{7}{10}y" } ], "meta": { "interimType": "Generic Simplify Specific 1Eq" } }, { "type": "interim", "title": "Simplify $$\\frac{7}{20}-\\frac{2}{5}y-\\frac{3}{4}:{\\quad}-\\frac{2\\left(y+1\\right)}{5}$$", "input": "\\frac{7}{20}-\\frac{2}{5}y-\\frac{3}{4}", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=-\\frac{2}{5}y-\\frac{3}{4}+\\frac{7}{20}" }, { "type": "interim", "title": "Multiply $$\\frac{2}{5}y\\::{\\quad}\\frac{2y}{5}$$", "input": "\\frac{2}{5}y", "result": "=-\\frac{2y}{5}-\\frac{3}{4}+\\frac{7}{20}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{2y}{5}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "interim", "title": "Least Common Multiplier of $$5,\\:4,\\:20:{\\quad}20$$", "input": "5,\\:4,\\:20", "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 $$5:{\\quad}5$$", "input": "5", "steps": [ { "type": "step", "primary": "$$5$$ is a prime number, therefore no factorization is possible", "result": "=5" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRjl/dE9e0owjU0NK6lxSAv4/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp3mWpvzkJh0pk9SzVPr3Sj8" } }, { "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 $$20:{\\quad}2\\cdot\\:2\\cdot\\:5$$", "input": "20", "steps": [ { "type": "step", "primary": "$$20\\:$$divides by $$2\\quad\\:20=10\\cdot\\:2$$", "result": "=2\\cdot\\:10" }, { "type": "step", "primary": "$$10\\:$$divides by $$2\\quad\\:10=5\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:5" }, { "type": "step", "primary": "$$2,\\:5$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:2\\cdot\\:5" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRl8ZboA8wPLg0yhI4RzfjFzwPjJo8Fi+uo29zEltAtWEB4gitN/2ICkrV6ivfiR3BLFRzd4QlsM8ugKm4vxBIEDIX3G4SX/N77p2fbACOqjb" } }, { "type": "step", "primary": "Compute a number comprised of factors that appear in at least one of the following:<br/>$$5,\\:4,\\:20$$", "result": "=5\\cdot\\:2\\cdot\\:2" }, { "type": "step", "primary": "Multiply the numbers: $$5\\cdot\\:2\\cdot\\:2=20$$", "result": "=20" } ], "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 $$20$$" }, { "type": "step", "primary": "For $$\\frac{2y}{5}:\\:$$multiply the denominator and numerator by $$4$$", "result": "\\frac{2y}{5}=\\frac{2y\\cdot\\:4}{5\\cdot\\:4}=\\frac{8y}{20}" }, { "type": "step", "primary": "For $$\\frac{3}{4}:\\:$$multiply the denominator and numerator by $$5$$", "result": "\\frac{3}{4}=\\frac{3\\cdot\\:5}{4\\cdot\\:5}=\\frac{15}{20}" } ], "meta": { "interimType": "LCD Adjust Fractions 1Eq" } }, { "type": "step", "result": "=-\\frac{8y}{20}-\\frac{15}{20}+\\frac{7}{20}" }, { "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{-8y-15+7}{20}" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-15+7=-8$$", "result": "=\\frac{-8y-8}{20}" }, { "type": "interim", "title": "Factor $$-8y-8:{\\quad}-8\\left(y+1\\right)$$", "input": "-8y-8", "result": "=-\\frac{8\\left(y+1\\right)}{20}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=-8y-8\\cdot\\:1" }, { "type": "step", "primary": "Factor out common term $$8$$", "result": "=-8\\left(y+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$4$$", "result": "=-\\frac{2\\left(y+1\\right)}{5}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mqPjITQWKpgVJEHsd+bLNY8/1FCg5Ck2gfaAJcql18fnABjRbRe4X0Y0IN18vpcdzRqDxPUzBN6vjj5oJL9kUNZ8zfti/Eio9CoKo8SaXqKePkiM5SOIVQuW1vlBOvIa1sD7NfhsPe7eDHrmjY0mE8DwQcN28geUT7aUsumC7dol8aXzjEsvpYVNdwOWhuUUWbGDFPi9iLyTTlzl0v3PaLsav6oKwOalk3ENSmN9wNU=" } }, { "type": "step", "result": "\\frac{7}{10}y=-\\frac{2\\left(y+1\\right)}{5}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Multiply by LCM", "input": "\\frac{7}{10}y=-\\frac{2\\left(y+1\\right)}{5}", "result": "7y=-4\\left(y+1\\right)", "steps": [ { "type": "interim", "title": "Find Least Common Multiplier of $$10,\\:5:{\\quad}10$$", "input": "10,\\:5", "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 $$10:{\\quad}2\\cdot\\:5$$", "input": "10", "steps": [ { "type": "step", "primary": "$$10\\:$$divides by $$2\\quad\\:10=5\\cdot\\:2$$", "result": "=2\\cdot\\:5" }, { "type": "step", "primary": "$$2,\\:5$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:5" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRvTIRluRNPwUULD5JCqpmdzwPjJo8Fi+uo29zEltAtWEB4gitN/2ICkrV6ivfiR3BLFRzd4QlsM8ugKm4vxBIEBeUcq3/p8vCxgx1MIzcdLE" } }, { "type": "interim", "title": "Prime factorization of $$5:{\\quad}5$$", "input": "5", "steps": [ { "type": "step", "primary": "$$5$$ is a prime number, therefore no factorization is possible", "result": "=5" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRjl/dE9e0owjU0NK6lxSAv4/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp3mWpvzkJh0pk9SzVPr3Sj8" } }, { "type": "step", "primary": "Multiply each factor the greatest number of times it occurs in either $$10$$ or $$5$$", "result": "=2\\cdot\\:5" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:5=10$$", "result": "=10" } ], "meta": { "solvingClass": "LCM", "interimType": "LCM Top in Equation Title 1Eq" } }, { "type": "step", "primary": "Multiply by LCM=$$10$$", "result": "\\frac{7}{10}y\\cdot\\:10=-\\frac{2\\left(y+1\\right)}{5}\\cdot\\:10" }, { "type": "interim", "title": "Simplify", "input": "\\frac{7}{10}y\\cdot\\:10=-\\frac{2\\left(y+1\\right)}{5}\\cdot\\:10", "result": "7y=-4\\left(y+1\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{7}{10}y\\cdot\\:10:{\\quad}7y$$", "input": "\\frac{7}{10}y\\cdot\\:10", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{7\\cdot\\:10}{10}y" }, { "type": "step", "primary": "Cancel the common factor: $$10$$", "result": "=y\\cdot\\:7" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7DzvLoUlgZkEr5z32+GDFI3XKCt5o3SVeM5shS8buwrIDnzlbPZjyKgy1eUCFsLd5e/Ub0Dx1YdX+JhxvdjMNMGRLd2VwIqlBNByF6663syR2SpdpleAJc7YgKUwBYoM9l0aiPa5kujgm90GLLvBZXRkHYhobNPyOnMHSBmVbmxI=" } }, { "type": "interim", "title": "Simplify $$-\\frac{2\\left(y+1\\right)}{5}\\cdot\\:10:{\\quad}-4\\left(y+1\\right)$$", "input": "-\\frac{2\\left(y+1\\right)}{5}\\cdot\\:10", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-\\frac{2\\left(y+1\\right)\\cdot\\:10}{5}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:10=20$$", "result": "=-\\frac{20\\left(y+1\\right)}{5}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{20}{5}=4$$", "result": "=-4\\left(y+1\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Mx0EtBP0Hqjp/bywLouvBajf5yq+HW6uVf1czit7ku/ehkKrn0era9rz8TlL+x/vzFhtOIr1n4Gj+qebzswjUe5byrQDQVCXUD0vH/fvOdxyhd7tjiG+GxQNxDvGkZUlZNh4AHfHcwyI5fJwyv4NYlZ/nxnZ/1ZT5Au1MziFxDG/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "result": "7y=-4\\left(y+1\\right)" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Equation LCM Multiply Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjAjYcBSbJJKcO5nwp2W+NzB2KfigobqBiR5xD74laQ0uCjUIutL/kO5uU2VX7H/4uB1w4NWdymWGg64uL7J8/NOw7w67GzocKku/Cy2jPb4KFcGppVvw1FZl7uHFH2U8bvIMhJPlSecmxOWh2YGNwSeZhAG/r3qz/Vtit5/x/2nfw==" } }, { "type": "interim", "title": "Expand $$-4\\left(y+1\\right):{\\quad}-4y-4$$", "input": "-4\\left(y+1\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=-4,\\:b=y,\\:c=1$$" ], "result": "=-4y+\\left(-4\\right)\\cdot\\:1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=-4y-4\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1=4$$", "result": "=-4y-4" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/xngS86rbX++SuD0mS5Q1XcUJadsLvGcDY7IUPYXjf6zs903yhxK0NInTwR7JvSRyM/29l7SHlwNleTQoRn0qBJyf8zawtgEaDEKWrMLEzacLZDOmB3jxgVx0Ieux+VX" } }, { "type": "step", "result": "7y=-4y-4" }, { "type": "interim", "title": "Move $$4y\\:$$to the left side", "input": "7y=-4y-4", "result": "11y=-4", "steps": [ { "type": "step", "primary": "Add $$4y$$ to both sides", "result": "7y+4y=-4y-4+4y" }, { "type": "step", "primary": "Simplify", "result": "11y=-4" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7R40M7n2+0qBV08N+hp4vAldr/F7DJva66mBn1mNE5axbx/37/XmD6my6awSUpo5T88UxTzlGSTEJ2AYfegcR+3jytKMkvdc+GZ2rr88qeA6tMfoIfYB+jIoSvT79TF75xmjGSXtcoI1/PQEtNvwvV90Wljz5MlBRGHaUS3SDbfg9ZP8VylC5Gq2Wd1dHmEckTwpiA4YlqqElAMsfEHHlYfVpjm0+Y86rif397S1JyPailM99yM3fjecbKwDRB7Xr5FOU5DDFhEudchbE5CU+pX5QeOPnAIW5dOQ+zdaOF52FA8Qbs7VnVnRJjlwXtyy6VjzasxIZD+ag+oqDVOLPX4vXP5xxoSKrz0abbXUotpaSY/+cSinQeRR8DuBv3x5/JsReacQHo4DAXV3/DR5fppN3SvYTeqT5M1TX8/JI3LeP4vUCm++DwNLv4kI5vGdyk7uyN8Yjotp1h/wFPXDFuLCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "Divide both sides by $$11$$", "input": "11y=-4", "result": "y=-\\frac{4}{11}", "steps": [ { "type": "step", "primary": "Divide both sides by $$11$$", "result": "\\frac{11y}{11}=\\frac{-4}{11}" }, { "type": "step", "primary": "Simplify", "result": "y=-\\frac{4}{11}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": 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