.pf{position:relative;background-color:#fff;overflow:hidden;margin:0;border:0}.pc{position:absolute;border:0;padding:0;margin:0;top:0;left:0;width:100%;height:100%;overflow:hidden;display:block;transform-origin:0 0;-ms-transform-origin:0 0;-webkit-transform-origin:0 0}.bi{position:absolute;border:0;margin:0}.c{position:absolute;border:0;padding:0;margin:0;overflow:hidden;display:block}.t{position:absolute;white-space:pre;font-size:1px;transform-origin:0 100%;-ms-transform-origin:0 100%;-webkit-transform-origin:0 100%;unicode-bidi:bidi-override;-moz-font-feature-settings:"liga" 0}.t:after{content:''}.t:before{content:'';display:inline-block}.t span{position:relative;unicode-bidi:bidi-override}._{display:inline-block;color:transparent;z-index:-1}.pi{display:none}@media screen{.pf{margin:13px auto;box-shadow:1px 1px 3px 1px #333;border-collapse:separate}}@font-face{font-family:ff1;src:url(data:application/font-woff;base64,d09GRgABAAAAAA7wAA4AAAAAFPgAAQAAAAAAAAAAAAAAAAAAAAAAAAAAAABGRlRNAAAO1AAAABwAAAAcWqih10dERUYAAA64AAAAHAAAAB4AJwA0T1MvMgAAAbgAAABBAAAAVlWqXz1jbWFwAAACkAAAAMgAAAGyEWL8OWN2dCAAAANYAAAABAAAAAQAIQJ5Z2FzcAAADrAAAAAIAAAACP//AANnbHlmAAADvAAACYgAAA1s7aOaVGhlYWQAAAFEAAAANgAAADboebBVaGhlYQAAAXwAAAAgAAAAJAXqAqlobXR4AAAB/AAAAJMAAAC4Wc8HWGxvY2EAAANcAAAAXgAAAF5LgkgobWF4cAAAAZwAAAAcAAAAIAJQAHNuYW1lAAANRAAAAQAAAAIiXHBCtnBvc3QAAA5EAAAAbAAAAJGfoj8TAAEAAAABAADJwV9pXw889QAfA+gAAAAAtkNgAAAAAADP9w1m//f/LwMVAtsAAAAIAAIAAAAAAAB4nGNgZGBguv1fn4GBOfT/9//fmEUZgCIoQA8AoRcGl3icY2BkYGDQY3BmYGYAASYwKQdm6IGYAA5LAMZ4nGNgZGJjnMDAysDA1MW0h4GBoQdCMz5gMGRkAooysDIzwAAjAxIISHNNAVIKDFVMt//rA1XeZrgIUwMAspUKkwAAAHicY8xhUGQAAkZfIBYDYmYGfUbm/9+ZIhgMgPxgEGbSYZAF8Zk6/v9iWsvgx3SJQZspGUjbAfF3IFsHLOYL5DMBxRmYJBlUmAIZnKG0Cohm1GDgAcqrANU6MNxjcAVh5lAGBxAfLA5UD3SDLeMXBnlGawZOkDiQzcf0jUGQiY2BE8jmYHzAIAakjZnYGVwBZnkdbAB4nGNgYGBmgGAZBkYGEFgD5DGC+SwME4C0AhCygGlNBh0GPQZjBlsGZwZ3Bh8GP4YQhjCGTIYChqr//8EqNBi0oSocGVzhKhIZchiK/v////j/nf+3/9/6f+3/2f+n/p/8f/T/kf/7/+/7v+X/pv8bobbjBYxsDHBljExAggldAcQrIMDCysDGzsABYnLC5bmAmJuHl4GPXwDEFWQQAgsLM4ggzBAVE5eQlJKWkWVgkJNXUFRiUFZRVVPX0NTSJuxAOgAAod8siAAhAnkAAAAqACoAKgAqAEYAYgB4AJAAnADwAQIBIAFkAZoBsgHIAggCGAIyAkQCWAK0AvYDKgNqA6YDygQeBEgEWgRoBKYEzAUGBUYFYAWwBdQF+gYOBi4GSAZwBoYGnAa2AAB4nJ1Xa3AbVxnde2VJtpxIXkkrybZeq5W0elmv9eq1etnxQ7JixXaixHYebhMndqGuZ5I0lDSTlseElpkw6Y9QmswAQ3nMAIVJoAUylGEKmZSQ8oNh2mmSH9BAB5hQMjBtmQat+XZXShxPfyFZ0t3d77v3O+eee+41gQmaIFAInyNUhJaIXkBELH9R20H8I3lBo76Zv6jC0CQuqKTbaun2Ra0G/Td/EUn3OZImWY5kaNT11ytX8LnmIRrPQndEbE1Am/ENwkAQVi6K+MFUmtEjyuzE14bGC/nKUH2bJZi7fOSdk0/+8fDSnWPDTxxehby1D9cE4g9KnpYrQl4Us4wTEjVdZTnFWx4r5C8v/RMyjqxmjtw6efxP0nhxgsAB9BHRDRc8QzEUR3E8x6P6yPXrI9eR68aNEfgjCCl2+9pjxGvEKrGJINIUZ9FjLVVEBbRdQGq9lcwNrq5usvdbNUlyJntQjocsKV4lx/PbV+El3XesvY9EfJkoEoSJSnFJJ6bMesx4/CyjcWIuWcT8oJ/lFRSMB8bhBotw3yKFhZGH8Vy39bvTAVtvKOOqbsUNRBdnU8m5YbY0FMrQuW1RfjrlwI0emvOkKhZfot+oVnegsxUVyXC0m2NMu0bTOwVnf2qaL+/TY9NEPFGJWj35HZwrE+rNZu0Rh4HAwM0x4Obz8vwCArqAaSqO/iN2XhoZOTUikYLXPoKvM/gqxACD/YgzqRCXpq0c+s2n8ae+J/794SkUrT8i/g2xaFL8EXqxmXrrLQiHl4qoQ+7TcKEDJUWAC16hQsMgkl5Pgp+NqhjVvacv/cKVDtnGp8VbaI6fTtvHhrl0gukZGOSt33zbmQn3jZfQ87e0tog/m0VcM8VkJ9jhPXps3p5NTVA6nRo1T33Y3R9l8q15Da/dwd8F7fjggoOhgHFJcFxS0h5cRqEUmX88stA5WF/MzDzViARnTmx/9qv6BZWTr8bKewS7Xdhdrq2gf2d35ZzCoWfr9WcWs1/+QmRr2hWfXilklqfjW2EsCXMRMANbJpLm2+86uiOuoiPiafQlfLWZmvpw6taUXJsUz0B8lxTPtTLqC6gi/hQdk0LRuBLawvEzwAFc+qwQK2v0fvlObH0QHV7szYtX/oydws5MfnfRbRf2DgXGM37tgi49vZRrPNUIR2dPTpVOzM2hJ6YSjZI3tm0pk11pDJr8QrA0l+svfeJ0feyzi0LiXq0s1KpuYSPr6G3xlIwHH4fnk/D8VBs7R3IIDIBk0OQ8Ro8vz2PxGSkUF5q/gt9X8Ra5Tww5BcjRtnJ4mrqz8F7zMj7ePIWPT01JMZIQDyoxtIkBqTAmtCQe+B2+dugudHUWf7KZghDCu/Y+1mESdD0MDGlgwfF61OYnXVQpy02jZYoqWWuk2cIl06xerfAFisQ6C0uzzN4UU4r2OxMlminF7RQdNKcqqgb25icjzGjGozV3G77YM5jNRR1kv9ccyvuNeLMvFPL1eNJsJMMYNVrt5l6b3WPUBLPx4aBR58oMiB847eorm7q1XWafm3IYO61M0ChDI7ZA3QOAzyavEuCAc6oUl2jVSyLzgxO9pYH1wWjMJMwJDqcwl0/NUxgJPV4hEimyRtKfDwcLrEmie9TqMnaGqvvT6f21CBtG3WLBX8l46NS43zuW9rhTo4q2gDv0EXDHfNwaQevWCPrarNovTEYKewpOV2G3cOCIflfneCmQ85I9vmI0VUYL0S0RKlw7mMs9POZffig/7OZHvGw140m15kgPWE0S1nQLDtWCp5b08uDgHHWyx5sPr4eFzsjoo+b16MUNqK7jq+IHVpepMyxjn4j4w2NtjagAp47wA1JaUYK1BTFNpxRDpqQJSKHviG9hi5+n3Txr3bGjezQVKgaMCH0OU+ndo/zckBe7iruLs0fRoJMPWK1s6gdc0h4reGLLs9lAZb8gLFYCswq/sKegIP49YW3rnIGtD+i08lEYWuNADHWgsbjocXX3dW9ybRqr7kKXxDF0qVbx2FQdEx0dQ6XJmlx/DPYWkggQedA4CTualL+erxaXsDtKGNh1qpEeIXEKWWcSW/ZkbI7czlxyzoJJfy5ERdl+1MCBQs174uh7oYxHT/pywUCG6SFBVOjtiXCY2/ZwlF8YDw3EPeEK5+gyu62hnLfn5FeY7BjjGxl0uflhj7eSZWS8Zfi6DfPcp+Bt7W9Sq7XlMWS5YfJn2USmQYZGOGtiwKvDsuF5GSFiy8TFb6AdodGkXUfRNkRKcydJ9Zy8F0luwTMOBP2dazTw/uXlvc27WA3j3o+Rxx1tyJ5ztV2TigLuQutqkmwAmql7LWXJSaRqGb7cMNC8z+GzdDUqJRdr6Wz0DpRYrtHXU4+v5DBWN++iYcNAyEm6Qr3iS2g4XzG6QjZoPRnIesloKB6Y3bqOD7I9/xv4kIdax8YRXul7AxX7Zoi2hvugLxPBQm8b1hDb9rIW0Sd7IwJNC5He9u+8u7yvUNxbdrvLe4uFfWU3wrFqsq8vWY3Fqom+vkQ1lt1fDQar+7PZA9VQqHpA8SgBPIq851HkRo/i1f+fR6lhxnUPmtTYxsUM/BWAv0lsIvSKQ0pj8H5p2ELDHKlwo/MumbF3k5Mp+zzOl1YgxwWe9ibUnAOnYe6XC6eNj9kwO9oEHpWMvh5mt5VYR6zgdubjLrObNVFBjwU3VJ7sRMQzmmaStdla0uaLmPsSrO3F+Jag0eAvxnxJ2gzeb3JYzL0GTZeZ7o2VfAaSybDJjIukPLTN1aPRWVnFE2DPRw78dVkTytmUhMkT4Jwq0QgoX2nMzuodMXqINffq1UtYff58TfyJN2Lrqql0RgMaqinaApy3Qf022B/BQZWzJi+12p6Abq9XV3ZHP5Al3lTEhfaJ3wdx9Q/EZ6Av4Be9C32t32uDr86dxs8t/BJyptEPm3elMaUF+ReIM9yLk2M5eOPyt2aP4kf3LuPH9317zzJ+BPLmwEvhA5I+Lx5sY29CvnRWUHFpK6dKM1qW+e1rS2fOrLx+8dBnTqygHvHOm28i63svv6zw1bmWkHFaCEItndvg+OhEALCEVNK4HOo2OJiw3eDUdzt0fUwX88ZDz+Pn5kf9g7ShQ11Xa3utuzCW6lD6g37QVeivSz798jSCwwdtQT8WX0F58THkHRqqoadzNXnNcQRC1/DrivcgmqThBESiF8Q3EMeJS/APzln8QjOLrxCKB2HtvXMQnzbJ4MCLpi48evONw7clS/r1hPgvVP35O3It/wN8iLGReJylj8FqwkAQhr/VaClIoeChx5yKVLRJTuKlBcHooVYK2ltBZAmBkEBM7Jv0VfpIfY3+0T32UHBhdr6d+WdnBujxhaE5hj4Pjltc8eq4zSPfjj365tZxh555dtxV/FNK410rMjhVNdzihifHbT54d+xJ8+O4w525d9xlYF5YEjNjzUZ+yAJLxlF3RcqeHStxLWMZz9abeLiw2dFW6X63srWib0olEmSSlnrapM52gjkFuT5pfCmFxSdiTCA/lf237VkdMmIii1QfipgXeTUvysT60Tjwp/5fwykcjiajKAhVcOGaW6VKDhI3azUDnVdha8tDWuR+qDEu7fILRaRZhnicbca5DgEBAEDB2dUQQjRqBYLCsY3Vu+8jbgqF0v+XiChN8pIn9PWKRf5pfAqEEtIysnLyCopKyiqqauqa2jq6evoGhkbGJqZmllbWNrZ29g6OTs4urm5BGCSSi/vzMX80WqnfRPEbCAQQhwAAAAH//wACeJxjYGRgYOABYjEgZmJgBEJdIGYB8xgABY0AXAAAAAEAAAAA1G40cAAAAAC2Q2AAAAAAAM/3DWY=) format("woff")}.ff1{font-family:ff1;line-height:.94;font-style:normal;font-weight:400;visibility:visible}@font-face{font-family:ff2;src:url(data:application/font-woff;base64,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) format("woff")}.ff2{font-family:ff2;line-height:.712;font-style:normal;font-weight:400;visibility:visible}.m0{transform:matrix(.320260,0,0,.320260,0,0);-ms-transform:matrix(.320260,0,0,.320260,0,0);-webkit-transform:matrix(.320260,0,0,.320260,0,0)}.ls0{letter-spacing:0}.sc0{text-shadow:-.015em 0 transparent,0 .015em transparent,.015em 0 transparent,0 -.015em transparent}@media screen and (-webkit-min-device-pixel-ratio:0){.sc0{-webkit-text-stroke:.015em transparent;text-shadow:none}}.ws0{word-spacing:-.034558px}.ws1{word-spacing:0}._2{margin-left:-1179.565947px}._3{margin-left:-957.151086px}._4{margin-left:-619.036264px}._0{margin-left:-3.835931px}._1{margin-left:-1.900686px}.fc0{color:#000}.fs0{font-size:34.557934px}.y3e{bottom:-387.066783px}.y3d{bottom:-373.785738px}.y3c{bottom:-360.504693px}.y3b{bottom:-348.330402px}.y3a{bottom:-335.049357px}.y39{bottom:-321.768313px}.y38{bottom:-308.487268px}.y37{bottom:-295.206223px}.y36{bottom:-281.925178px}.y35{bottom:-268.644133px}.y34{bottom:-255.363089px}.y33{bottom:-242.082044px}.y32{bottom:-228.800999px}.y31{bottom:-215.519954px}.y30{bottom:-202.23891px}.y2f{bottom:-188.957865px}.y2e{bottom:-175.67682px}.y2d{bottom:-163.502529px}.y2c{bottom:-150.221484px}.y2b{bottom:-136.940439px}.y2a{bottom:-123.659394px}.y29{bottom:-110.37835px}.y28{bottom:-97.097305px}.y27{bottom:-83.81626px}.y26{bottom:-70.535215px}.y25{bottom:-57.254171px}.y24{bottom:-43.973126px}.y23{bottom:-30.692081px}.y22{bottom:-17.411036px}.y21{bottom:-4.129991px}.y0{bottom:0}.y20{bottom:9.151053px}.y1f{bottom:22.432098px}.y1e{bottom:34.606389px}.y1d{bottom:47.887434px}.y1c{bottom:61.168479px}.y1b{bottom:74.449524px}.y1a{bottom:87.730568px}.y19{bottom:101.011613px}.y18{bottom:114.292658px}.y17{bottom:127.573703px}.y16{bottom:140.854748px}.y15{bottom:154.135792px}.y14{bottom:167.416837px}.y13{bottom:180.697882px}.y12{bottom:193.978927px}.y11{bottom:207.259972px}.y10{bottom:219.434263px}.yf{bottom:232.715307px}.ye{bottom:245.996352px}.yd{bottom:259.277397px}.yc{bottom:272.558442px}.yb{bottom:285.839487px}.ya{bottom:299.120531px}.y9{bottom:312.401576px}.y8{bottom:325.682621px}.y7{bottom:338.963666px}.y6{bottom:352.244711px}.y5{bottom:365.525755px}.y4{bottom:378.806800px}.y3{bottom:392.087845px}.y2{bottom:404.262136px}.y1{bottom:507.293741px}.h4{height:24.605249px}.h3{height:25.261849px}.h2{height:507.295022px}.h1{height:909.539511px}.h0{height:1014.588763px}.w1{width:783.997438px}.w0{width:784px}.x0{left:0}.x1{left:91.953969px}.x2{left:103.021506px}

MAT 211 Lecture Notes - Lecture 9: Linear Function

United States

Mathematics for Business Analysis

Mathematics

Mohacsy Hedvig

Arizona State University

Bones, Stones, and Human Evolution

Macroeconomic Principles

Brief Calculus

Introductory Sociology

Interior Design Issues and Theories

Function of Two Variables
What is a graph of a function of two variables? How is it interpreted geometrically?

Systems of Linear Equations in Two Variables Solve the system of equations, and graph the lines.<img src="data:image/png;base64,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" class="fm-editor-equation" data-mlang="latex" data-equation="%7Bbegin%7Bmatrix%7D%0A3x-y%3D5%0A2x%2By%3D5%0Aend%7Bmatrix%7D.">

Systems of Linear Equations in Two Variables Solve the system of equations, and graph the lines.<img src="data:image/png;base64,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" class="fm-editor-equation" data-mlang="latex" data-equation="%7Bbegin%7Bmatrix%7D%0Ay%3D2x%2B6%0Ay%3D-x%2B3%0Aend%7Bmatrix%7D.">

MAT 211 Lecture Notes - Lecture 9: Linear Function

Get access

Related Documents

MAT 211 Lecture Notes - Lecture 11: Partial Derivative, Linear Function, Power Rule

MAT 211 Lecture Notes - Lecture 21: 5,6,7,8, Random Variable, Sample Space

MAT 211 Lecture Notes - Lecture 24: Random Variable, Standard Normal Deviate, Standard Deviation

Related Questions