{"id":355,"date":"2018-04-17T02:19:46","date_gmt":"2018-04-17T02:19:46","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/?post_type=chapter&#038;p=355"},"modified":"2024-04-26T22:04:43","modified_gmt":"2024-04-26T22:04:43","slug":"variables","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/chapter\/variables\/","title":{"raw":"Variables","rendered":"Variables"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Define and identify variables<\/li>\r\n<\/ul>\r\n<\/div>\r\n<img class=\"wp-image-830 alignright\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2985\/2018\/04\/30213939\/32381921555_c399a66856_o-300x225.jpg\" alt=\"Two men dressed similarly, with the same haircut, and both wear round glasses take a selfie with each other.\" width=\"298\" height=\"223\" \/>Greg and Alex have the same birthday, but they were born in different years. This year Greg is [latex]20[\/latex] years old and Alex is [latex]23[\/latex], so Alex is [latex]3[\/latex] years older than Greg. When Greg was [latex]12[\/latex], Alex was [latex]15[\/latex]. When Greg is [latex]35[\/latex], Alex will be [latex]38[\/latex]. No matter what Greg\u2019s age is, Alex\u2019s age will always be [latex]3[\/latex] years more, right?\r\n\r\nIn the language of algebra, we say that Greg\u2019s age and Alex\u2019s age are variable and the three is a constant. The ages change, or vary, so age is a variable. The [latex]3[\/latex] years between them always stays the same, so the age difference is the constant.\r\n\r\nIn mathematics, letters of the alphabet are used to represent variables. Suppose we call Greg\u2019s age [latex]g[\/latex]. Then we could use [latex]g+3[\/latex] to represent Alex\u2019s age. See the table below.\r\n<table style=\"width: 40%;\" summary=\"This table has five rows and two columns. The first row is a header row and is labeled \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th data-align=\"center\">Greg\u2019s age<\/th>\r\n<th data-align=\"center\">Alex\u2019s age<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"center\">[latex]12[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]15[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"center\">[latex]20[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]23[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"center\">[latex]35[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]38[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"center\">[latex]g[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]g+3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nLetters are used to represent variables. Letters often used for variables are [latex]x,y,a,b,\\text{ and }c[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>Variables and Constants<\/h3>\r\n<p style=\"padding-left: 30px;\">A variable is a letter that represents a number or quantity whose value may change (ex. [latex]x, y, z, a, t, k[\/latex] etc.).<\/p>\r\n<p style=\"padding-left: 30px;\">A constant is a number whose value always stays the same.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>EXAMPLE<\/h3>\r\nIdentify the variable(s) in each expression or equation\r\n<ol>\r\n \t<li>[latex]x+2[\/latex]<\/li>\r\n \t<li>[latex]5-3y[\/latex]<\/li>\r\n \t<li>[latex]7+5b-z=9[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"114888\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"114888\"]\r\n\r\nSolution\r\n<ol>\r\n \t<li>[latex]x[\/latex]<\/li>\r\n \t<li>[latex]y[\/latex]<\/li>\r\n \t<li>[latex]b[\/latex] and [latex]z[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY IT<\/h3>\r\n[ohm_question]156972[\/ohm_question]\r\n\r\n<\/div>\r\nTo write algebraically, we need some symbols as well as numbers and variables. There are several types of symbols we will be using. There are multiple symbols and phrases to represent the four basic arithmetic operations: addition, subtraction, multiplication, and division. We will summarize them here:\r\n<table class=\"unnumbered\" style=\"width: 40%;\" summary=\"This table has five rows and four columns. The first row is a header row. Each column is labeled accordingly: the first is labeled \" data-label=\"\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th data-align=\"left\"><strong>Operation<\/strong><\/th>\r\n<th data-align=\"left\"><strong>Notation<\/strong><\/th>\r\n<th data-align=\"left\"><strong>Say:<\/strong><\/th>\r\n<th data-align=\"left\"><strong>The result is\u2026<\/strong><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">Addition<\/td>\r\n<td data-align=\"left\">[latex]a+b[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]a\\text{ plus }b[\/latex]<\/td>\r\n<td data-align=\"left\">the sum of [latex]a[\/latex] and [latex]b[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">Subtraction<\/td>\r\n<td data-align=\"left\">[latex]a-b[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]a\\text{ minus }b[\/latex]<\/td>\r\n<td data-align=\"left\">the difference of [latex]a[\/latex] and [latex]b[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">Multiplication<\/td>\r\n<td data-align=\"left\">[latex]a\\cdot b,\\left(a\\right)\\left(b\\right),\\left(a\\right)b,a\\left(b\\right)[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]a\\text{ times }b[\/latex]<\/td>\r\n<td data-align=\"left\">The product of [latex]a[\/latex] and [latex]b[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">Division<\/td>\r\n<td data-align=\"left\">[latex]a\\div b,a\/b,\\frac{a}{b},b\\overline{)a}[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]a[\/latex] divided by [latex]b[\/latex]<\/td>\r\n<td data-align=\"left\">The quotient of [latex]a[\/latex] and [latex]b[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn algebra, the cross symbol, [latex]\\times [\/latex], is not used to show multiplication because that symbol may cause confusion. Does [latex]3xy[\/latex] mean [latex]3\\times y[\/latex] (three times [latex]y[\/latex] ) or [latex]3\\cdot x\\cdot y[\/latex] (three times [latex]x\\text{ times }y[\/latex] )? To make it clear, use \u2022 or parentheses for multiplication.\r\n\r\nGrouping symbols in algebra are much like the commas, colons, and other punctuation marks in written language. They indicate which expressions are to be kept together and separate from other expressions. The table below lists three of the most commonly used grouping symbols in algebra.\r\n<table style=\"width: 40%;\" summary=\"This table has four rows and two columns. The first row spans both columns and is a header reading \">\r\n<thead>\r\n<tr style=\"height: 15px;\" valign=\"top\">\r\n<th style=\"width: 93.6508%; height: 15px;\" colspan=\"2\" data-align=\"center\"><strong>Common Grouping Symbols<\/strong><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 15px;\" valign=\"top\">\r\n<td style=\"width: 61.4286%; height: 15px;\" data-align=\"left\">parentheses<\/td>\r\n<td style=\"width: 32.2222%; height: 15px;\" data-align=\"center\">( )<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"top\">\r\n<td style=\"width: 61.4286%; height: 15px;\" data-align=\"left\">brackets<\/td>\r\n<td style=\"width: 32.2222%; height: 15px;\" data-align=\"center\">[ ]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"top\">\r\n<td style=\"width: 61.4286%; height: 15px;\" data-align=\"left\">braces<\/td>\r\n<td style=\"width: 32.2222%; height: 15px;\" data-align=\"center\">{ }<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nHere are some examples of expressions that include grouping symbols.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}8\\left(14 - 8\\right)21 - 3\\\\\\left[2+4\\left(9 - 8\\right)\\right]\\\\24\\div \\left\\{13 - 2\\left[1\\left(6 - 5\\right)+4\\right]\\right\\}\\end{array}[\/latex]<\/p>\r\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Define and identify variables<\/li>\n<\/ul>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-830 alignright\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2985\/2018\/04\/30213939\/32381921555_c399a66856_o-300x225.jpg\" alt=\"Two men dressed similarly, with the same haircut, and both wear round glasses take a selfie with each other.\" width=\"298\" height=\"223\" \/>Greg and Alex have the same birthday, but they were born in different years. This year Greg is [latex]20[\/latex] years old and Alex is [latex]23[\/latex], so Alex is [latex]3[\/latex] years older than Greg. When Greg was [latex]12[\/latex], Alex was [latex]15[\/latex]. When Greg is [latex]35[\/latex], Alex will be [latex]38[\/latex]. No matter what Greg\u2019s age is, Alex\u2019s age will always be [latex]3[\/latex] years more, right?<\/p>\n<p>In the language of algebra, we say that Greg\u2019s age and Alex\u2019s age are variable and the three is a constant. The ages change, or vary, so age is a variable. The [latex]3[\/latex] years between them always stays the same, so the age difference is the constant.<\/p>\n<p>In mathematics, letters of the alphabet are used to represent variables. Suppose we call Greg\u2019s age [latex]g[\/latex]. Then we could use [latex]g+3[\/latex] to represent Alex\u2019s age. See the table below.<\/p>\n<table style=\"width: 40%;\" summary=\"This table has five rows and two columns. The first row is a header row and is labeled\">\n<thead>\n<tr valign=\"top\">\n<th data-align=\"center\">Greg\u2019s age<\/th>\n<th data-align=\"center\">Alex\u2019s age<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"center\">[latex]12[\/latex]<\/td>\n<td data-align=\"center\">[latex]15[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"center\">[latex]20[\/latex]<\/td>\n<td data-align=\"center\">[latex]23[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"center\">[latex]35[\/latex]<\/td>\n<td data-align=\"center\">[latex]38[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"center\">[latex]g[\/latex]<\/td>\n<td data-align=\"center\">[latex]g+3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Letters are used to represent variables. Letters often used for variables are [latex]x,y,a,b,\\text{ and }c[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>Variables and Constants<\/h3>\n<p style=\"padding-left: 30px;\">A variable is a letter that represents a number or quantity whose value may change (ex. [latex]x, y, z, a, t, k[\/latex] etc.).<\/p>\n<p style=\"padding-left: 30px;\">A constant is a number whose value always stays the same.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE<\/h3>\n<p>Identify the variable(s) in each expression or equation<\/p>\n<ol>\n<li>[latex]x+2[\/latex]<\/li>\n<li>[latex]5-3y[\/latex]<\/li>\n<li>[latex]7+5b-z=9[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q114888\">Show Answer<\/span><\/p>\n<div id=\"q114888\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<ol>\n<li>[latex]x[\/latex]<\/li>\n<li>[latex]y[\/latex]<\/li>\n<li>[latex]b[\/latex] and [latex]z[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>TRY IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm156972\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=156972&theme=oea&iframe_resize_id=ohm156972&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>To write algebraically, we need some symbols as well as numbers and variables. There are several types of symbols we will be using. There are multiple symbols and phrases to represent the four basic arithmetic operations: addition, subtraction, multiplication, and division. We will summarize them here:<\/p>\n<table class=\"unnumbered\" style=\"width: 40%;\" summary=\"This table has five rows and four columns. The first row is a header row. Each column is labeled accordingly: the first is labeled\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th data-align=\"left\"><strong>Operation<\/strong><\/th>\n<th data-align=\"left\"><strong>Notation<\/strong><\/th>\n<th data-align=\"left\"><strong>Say:<\/strong><\/th>\n<th data-align=\"left\"><strong>The result is\u2026<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"left\">Addition<\/td>\n<td data-align=\"left\">[latex]a+b[\/latex]<\/td>\n<td data-align=\"left\">[latex]a\\text{ plus }b[\/latex]<\/td>\n<td data-align=\"left\">the sum of [latex]a[\/latex] and [latex]b[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">Subtraction<\/td>\n<td data-align=\"left\">[latex]a-b[\/latex]<\/td>\n<td data-align=\"left\">[latex]a\\text{ minus }b[\/latex]<\/td>\n<td data-align=\"left\">the difference of [latex]a[\/latex] and [latex]b[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">Multiplication<\/td>\n<td data-align=\"left\">[latex]a\\cdot b,\\left(a\\right)\\left(b\\right),\\left(a\\right)b,a\\left(b\\right)[\/latex]<\/td>\n<td data-align=\"left\">[latex]a\\text{ times }b[\/latex]<\/td>\n<td data-align=\"left\">The product of [latex]a[\/latex] and [latex]b[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">Division<\/td>\n<td data-align=\"left\">[latex]a\\div b,a\/b,\\frac{a}{b},b\\overline{)a}[\/latex]<\/td>\n<td data-align=\"left\">[latex]a[\/latex] divided by [latex]b[\/latex]<\/td>\n<td data-align=\"left\">The quotient of [latex]a[\/latex] and [latex]b[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In algebra, the cross symbol, [latex]\\times[\/latex], is not used to show multiplication because that symbol may cause confusion. Does [latex]3xy[\/latex] mean [latex]3\\times y[\/latex] (three times [latex]y[\/latex] ) or [latex]3\\cdot x\\cdot y[\/latex] (three times [latex]x\\text{ times }y[\/latex] )? To make it clear, use \u2022 or parentheses for multiplication.<\/p>\n<p>Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in written language. They indicate which expressions are to be kept together and separate from other expressions. The table below lists three of the most commonly used grouping symbols in algebra.<\/p>\n<table style=\"width: 40%;\" summary=\"This table has four rows and two columns. The first row spans both columns and is a header reading\">\n<thead>\n<tr style=\"height: 15px;\" valign=\"top\">\n<th style=\"width: 93.6508%; height: 15px;\" colspan=\"2\" data-align=\"center\"><strong>Common Grouping Symbols<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 15px;\" valign=\"top\">\n<td style=\"width: 61.4286%; height: 15px;\" data-align=\"left\">parentheses<\/td>\n<td style=\"width: 32.2222%; height: 15px;\" data-align=\"center\">( )<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"top\">\n<td style=\"width: 61.4286%; height: 15px;\" data-align=\"left\">brackets<\/td>\n<td style=\"width: 32.2222%; height: 15px;\" data-align=\"center\">[ ]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"top\">\n<td style=\"width: 61.4286%; height: 15px;\" data-align=\"left\">braces<\/td>\n<td style=\"width: 32.2222%; height: 15px;\" data-align=\"center\">{ }<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Here are some examples of expressions that include grouping symbols.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}8\\left(14 - 8\\right)21 - 3\\\\\\left[2+4\\left(9 - 8\\right)\\right]\\\\24\\div \\left\\{13 - 2\\left[1\\left(6 - 5\\right)+4\\right]\\right\\}\\end{array}[\/latex]<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-355\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Brothers taking selfie. <strong>Authored by<\/strong>: Garry Knight. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.flickr.com\/photos\/8176740@N05\/32381921555\">https:\/\/www.flickr.com\/photos\/8176740@N05\/32381921555<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Brothers taking selfie\",\"author\":\"Garry Knight\",\"organization\":\"\",\"url\":\"https:\/\/www.flickr.com\/photos\/8176740@N05\/32381921555\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"}]","CANDELA_OUTCOMES_GUID":"8326a2db-5c35-4add-bcd8-f2f6fc41c836, dbff39f1-0727-4236-9944-09861c87bd0b","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-355","chapter","type-chapter","status-publish","hentry"],"part":26,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters\/355","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":14,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters\/355\/revisions"}],"predecessor-version":[{"id":3994,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters\/355\/revisions\/3994"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/parts\/26"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters\/355\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/media?parent=355"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapter-type?post=355"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/contributor?post=355"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/license?post=355"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}