{"id":5129,"date":"2023-06-26T18:54:19","date_gmt":"2023-06-26T18:54:19","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=5129"},"modified":"2024-10-18T20:53:29","modified_gmt":"2024-10-18T20:53:29","slug":"cryptography-background-youll-need-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/cryptography-background-youll-need-2\/","title":{"raw":"Cryptography: Background You\u2019ll Need 2","rendered":"Cryptography: Background You\u2019ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Substitute one or more numbers in for one or more variables and simplify<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Evaluate Multivariable Algebraic Expressions<\/h2>\r\nThe process for evaluating multivariable algebraic expressions is very similar to evaluating algebraic expressions with one variable. The only difference is that instead of substituting a number in for one variable, you will have to substitute one or more numbers in for multiple variables. The rest of the process is the same!\r\n\r\n<section class=\"textbox example\">Evaluate [latex]3x+4y - 6[\/latex] when [latex]x=10[\/latex] and [latex]y=2[\/latex].[reveal-answer q=\"412490\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"412490\"]This expression contains two variables, so we must make two substitutions.\r\n<table id=\"eip-id1168467158036\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression three x plus four y minus 6. Substitute 10 for x and 2 for y. The expression becomes 3 times 10 plus 4 times 2 minus 6. Perform multiplication from left to right. Three times 10 is 30 and 4 times 2 is 8. The expression becomes 30 plus 8 minus 6. Add and subtract from left to right. Thirty plus 8 is 38. Thirty-eight minus 6 is 32.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]3x+4y-6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{10}[\/latex] for [latex]x[\/latex] and [latex]\\color{blue}{2}[\/latex] for [latex]y[\/latex].<\/td>\r\n<td>[latex]3([\/latex][latex]\\color{red}{10}[\/latex][latex])+4([\/latex][latex]\\color{blue}{2}[\/latex][latex])-6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]30+8-6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add and subtract left to right.<\/td>\r\n<td>[latex]32[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhen [latex]x=10[\/latex] and [latex]y=2[\/latex], the expression [latex]3x+4y - 6[\/latex] has a value of [latex]32[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]8802[\/ohm2_question]<\/section><section class=\"textbox example\">Evaluate [latex]2{x}^{2}+3x+8[\/latex] when [latex]x=4[\/latex].[reveal-answer q=\"841995\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"841995\"]We need to be careful when an expression has a variable with an exponent. In this expression, [latex]2{x}^{2}[\/latex] means [latex]2\\cdot x\\cdot x[\/latex] and is different from the expression [latex]{\\left(2x\\right)}^{2}[\/latex], which means [latex]2x\\cdot 2x[\/latex].\r\n<table id=\"eip-id1168466011069\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression two x squared plus three x plus 8. Substitute 4 for each x. The expression becomes 2 times 4 squared plus 3 times 4 plus 8. Simplify exponents first. Four squared is 16 so the expression becomes 2 times 16 plus 3 times 4 plus 8. Next perform multiplication from left to right. Two times 16 is 32 and 3 times 4 is 12. The expression becomes 32 plus 12 plus 8. Add from left to right. Thirty-two plus 12 is 44. Forty-four plus 8 is 52.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]2x^2+3x+8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{4}[\/latex] for each [latex]x[\/latex].<\/td>\r\n<td>[latex]2([\/latex][latex]\\color{red}{4}[\/latex][latex])^{2}+3([\/latex][latex]\\color{red}{4}[\/latex][latex])+8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify [latex]{4}^{2}[\/latex] .<\/td>\r\n<td>[latex]2(16)+3(4)+8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]32+12+8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]52[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]8803[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Substitute one or more numbers in for one or more variables and simplify<\/li>\n<\/ul>\n<\/section>\n<h2>Evaluate Multivariable Algebraic Expressions<\/h2>\n<p>The process for evaluating multivariable algebraic expressions is very similar to evaluating algebraic expressions with one variable. The only difference is that instead of substituting a number in for one variable, you will have to substitute one or more numbers in for multiple variables. The rest of the process is the same!<\/p>\n<section class=\"textbox example\">Evaluate [latex]3x+4y - 6[\/latex] when [latex]x=10[\/latex] and [latex]y=2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q412490\">Show Answer<\/button><\/p>\n<div id=\"q412490\" class=\"hidden-answer\" style=\"display: none\">This expression contains two variables, so we must make two substitutions.<\/p>\n<table id=\"eip-id1168467158036\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression three x plus four y minus 6. Substitute 10 for x and 2 for y. The expression becomes 3 times 10 plus 4 times 2 minus 6. Perform multiplication from left to right. Three times 10 is 30 and 4 times 2 is 8. The expression becomes 30 plus 8 minus 6. Add and subtract from left to right. Thirty plus 8 is 38. Thirty-eight minus 6 is 32.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]3x+4y-6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{10}[\/latex] for [latex]x[\/latex] and [latex]\\color{blue}{2}[\/latex] for [latex]y[\/latex].<\/td>\n<td>[latex]3([\/latex][latex]\\color{red}{10}[\/latex][latex])+4([\/latex][latex]\\color{blue}{2}[\/latex][latex])-6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]30+8-6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add and subtract left to right.<\/td>\n<td>[latex]32[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>When [latex]x=10[\/latex] and [latex]y=2[\/latex], the expression [latex]3x+4y - 6[\/latex] has a value of [latex]32[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm8802\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=8802&theme=lumen&iframe_resize_id=ohm8802&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\">Evaluate [latex]2{x}^{2}+3x+8[\/latex] when [latex]x=4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q841995\">Show Answer<\/button><\/p>\n<div id=\"q841995\" class=\"hidden-answer\" style=\"display: none\">We need to be careful when an expression has a variable with an exponent. In this expression, [latex]2{x}^{2}[\/latex] means [latex]2\\cdot x\\cdot x[\/latex] and is different from the expression [latex]{\\left(2x\\right)}^{2}[\/latex], which means [latex]2x\\cdot 2x[\/latex].<\/p>\n<table id=\"eip-id1168466011069\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression two x squared plus three x plus 8. Substitute 4 for each x. The expression becomes 2 times 4 squared plus 3 times 4 plus 8. Simplify exponents first. Four squared is 16 so the expression becomes 2 times 16 plus 3 times 4 plus 8. Next perform multiplication from left to right. Two times 16 is 32 and 3 times 4 is 12. The expression becomes 32 plus 12 plus 8. Add from left to right. Thirty-two plus 12 is 44. Forty-four plus 8 is 52.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]2x^2+3x+8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{4}[\/latex] for each [latex]x[\/latex].<\/td>\n<td>[latex]2([\/latex][latex]\\color{red}{4}[\/latex][latex])^{2}+3([\/latex][latex]\\color{red}{4}[\/latex][latex])+8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify [latex]{4}^{2}[\/latex] .<\/td>\n<td>[latex]2(16)+3(4)+8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]32+12+8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]52[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm8803\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=8803&theme=lumen&iframe_resize_id=ohm8803&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":16,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"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\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":2245,"module-header":"background_you_need","content_attributions":[{"type":"cc-attribution","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"}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/5129"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/16"}],"version-history":[{"count":17,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/5129\/revisions"}],"predecessor-version":[{"id":12337,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/5129\/revisions\/12337"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/2245"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/5129\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=5129"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=5129"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=5129"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=5129"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}