simplify 5/(m-1)-2/(m-3)

{ "query": { "display": "simplify $$\\frac{5}{m-1}-\\frac{2}{m-3}$$", "symbolab_question": "SIMPLIFY#simplify \\frac{5}{m-1}-\\frac{2}{m-3}" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Algebra", "subTopic": "Simplify", "default": "\\frac{3m-13}{(m-1)(m-3)}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Simplify $$\\frac{5}{m-1}-\\frac{2}{m-3}:{\\quad}\\frac{3m-13}{\\left(m-1\\right)\\left(m-3\\right)}$$", "input": "\\frac{5}{m-1}-\\frac{2}{m-3}", "steps": [ { "type": "interim", "title": "Least Common Multiplier of $$m-1,\\:m-3:{\\quad}\\left(m-1\\right)\\left(m-3\\right)$$", "input": "m-1,\\:m-3", "steps": [ { "type": "definition", "title": "Lowest Common Multiplier (LCM)", "text": "The LCM of $$a,\\:b\\:$$is the smallest multiplier that is divisible by both $$a$$ and $$b$$" }, { "type": "step", "primary": "Compute an expression comprised of factors that appear either in $$m-1$$ or $$m-3$$", "result": "=\\left(m-1\\right)\\left(m-3\\right)" } ], "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 $$\\left(m-1\\right)\\left(m-3\\right)$$" }, { "type": "step", "primary": "For $$\\frac{5}{m-1}:\\:$$multiply the denominator and numerator by $$m-3$$", "result": "\\frac{5}{m-1}=\\frac{5\\left(m-3\\right)}{\\left(m-1\\right)\\left(m-3\\right)}=\\frac{5\\left(m-3\\right)}{\\left(m-1\\right)\\left(m-3\\right)}" }, { "type": "step", "primary": "For $$\\frac{2}{m-3}:\\:$$multiply the denominator and numerator by $$m-1$$", "result": "\\frac{2}{m-3}=\\frac{2\\left(m-1\\right)}{\\left(m-3\\right)\\left(m-1\\right)}=\\frac{2\\left(m-1\\right)}{\\left(m-1\\right)\\left(m-3\\right)}" } ], "meta": { "interimType": "LCD Adjust Fractions 1Eq" } }, { "type": "step", "result": "=\\frac{5\\left(m-3\\right)}{\\left(m-1\\right)\\left(m-3\\right)}-\\frac{2\\left(m-1\\right)}{\\left(m-1\\right)\\left(m-3\\right)}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{c}-\\frac{b}{c}=\\frac{a-b}{c}$$", "result": "=\\frac{5\\left(m-3\\right)-2\\left(m-1\\right)}{\\left(m-1\\right)\\left(m-3\\right)}" }, { "type": "interim", "title": "Simplify $$5\\left(m-3\\right)-2\\left(m-1\\right):{\\quad}3m-13$$", "input": "5\\left(m-3\\right)-2\\left(m-1\\right)", "steps": [ { "type": "interim", "title": "Expand $$5\\left(m-3\\right):{\\quad}5m-15$$", "input": "5\\left(m-3\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$5\\left(m-3\\right)=5m-5\\cdot\\:3$$" ], "result": "=5m-5\\cdot\\:3", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$5\\cdot\\:3=15$$", "result": "=5m-15" } ], "meta": { "solvingClass": "Solver2", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GNMKNOPqVJrXiIPh3rQ6ewsBmUOGgB0VAXoAXik9qC9EoApuujwRq/Ms3wa4gNT89ZVSO0B13oSpK+zJfqv7kB7MWKURN+43KCOzRc+RXaGVZlspPUxChIgF+pk9TvRC" } }, { "type": "step", "result": "=5m-15-2\\left(m-1\\right)" }, { "type": "interim", "title": "Expand $$-2\\left(m-1\\right):{\\quad}-2m+2$$", "input": "-2\\left(m-1\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$-2\\left(m-1\\right)=-2m-\\left(-2\\right)\\cdot\\:1$$" ], "result": "=-2m-\\left(-2\\right)\\cdot\\:1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "interim", "title": "$$-\\left(-2\\right)\\cdot\\:1=2$$", "input": "-\\left(-2\\right)\\cdot\\:1", "steps": [ { "type": "step", "primary": "Apply rule: $$-\\left(-a\\right)=a$$", "secondary": [ "$$-\\left(-2\\right)=2$$" ], "result": "=2\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zA1dH5zWL1siW/PXCNQxai061ljBSPJeENOw2efoSWt8rNweSBKaNqhWM5iGGSVMMb1QS31aBvYHwKLXjUkAb3lP9vXoXSyY9eaysz9yDlg=" } }, { "type": "step", "result": "=-2m+2" } ], "meta": { "solvingClass": "Solver2", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7kI5yzuMMD9CbUkSPAp272HcUJadsLvGcDY7IUPYXjf6zs903yhxK0NInTwR7JvSRyM/29l7SHlwNleTQoRn0qBJyf8zawtgEaDEKWrMLEzbikg9EbeAHs08wFjSvK/am" } }, { "type": "step", "result": "=5m-15-2m+2" }, { "type": "interim", "title": "Simplify $$5m-15-2m+2:{\\quad}3m-13$$", "input": "5m-15-2m+2", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=5m-2m-15+2" }, { "type": "step", "primary": "Add similar elements: $$5m-2m=3m$$", "result": "=3m-15+2" }, { "type": "step", "primary": "Add the numbers: $$-15+2=-13$$", "result": "=3m-13" } ], "meta": { "solvingClass": "Solver2", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7SM58qNN4+ObYFUGgiwqqJSAn9lkDfZkicUGkO3EF+IoMYu+sRrm2PHcz2KiorrPPo3oe/oyhMy2+1TQhDBd2f2zM6E3fuZxF1XkKAYaRXCBb8nlb5lqVVpZx49liHaQF" } }, { "type": "step", "result": "=3m-13" } ], "meta": { "solvingClass": "Solver2", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OoFITXqmTqi3MH+QMc3vRS061ljBSPJeENOw2efoSWtbzA/qN7RDXwVwFsb/E2pHLFkLZJTec9/y4HD02wKt2XKF3u2OIb4bFA3EO8aRlSUELqs+pNqtmW5BeRyK9TTe4gBJl4WMO1rA0a30/bUYlg==" } }, { "type": "step", "result": "=\\frac{3m-13}{\\left(m-1\\right)\\left(m-3\\right)}" } ], "meta": { "solvingClass": "Solver2" } }, "meta": { "showVerify": true } }