{"id":2644,"date":"2025-08-13T18:20:53","date_gmt":"2025-08-13T18:20:53","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2644"},"modified":"2025-09-16T18:05:16","modified_gmt":"2025-09-16T18:05:16","slug":"systems-of-equations-and-inequalities-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/systems-of-equations-and-inequalities-background-youll-need-1\/","title":{"raw":"Systems of Equations and Inequalities: Background You'll Need 1","rendered":"Systems of Equations and Inequalities: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Recognize when a linear equation has no solution or infinite solutions<\/span><\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>no solution \/ infinite solutions<\/h3>\r\n<p data-start=\"328\" data-end=\"434\">When solving a linear equation, sometimes the variable cancels out. What remains tells you the number of solutions:<\/p>\r\n\r\n<ul data-start=\"436\" data-end=\"691\">\r\n \t<li data-start=\"436\" data-end=\"537\">\r\n<p data-start=\"438\" data-end=\"537\">If you reach a true statement like [latex]0=0[\/latex], there are <strong data-start=\"507\" data-end=\"536\">infinitely many solutions<\/strong>.<\/p>\r\n<\/li>\r\n \t<li data-start=\"538\" data-end=\"625\">\r\n<p data-start=\"540\" data-end=\"625\">If you reach a false statement like [latex]0=5[\/latex], there is <strong data-start=\"609\" data-end=\"624\">no solution<\/strong>.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p data-start=\"705\" data-end=\"749\">Decide how many solutions each equation has.<\/p>\r\n\r\n<ol data-start=\"751\" data-end=\"1111\">\r\n \t<li data-start=\"751\" data-end=\"900\">\r\n<p data-start=\"754\" data-end=\"900\">[latex]3x+6 = 3(x+2)[\/latex]<\/p>\r\n[latex]\\begin{align}\r\n3x+6 &amp;= 3(x+2) \\\\\r\n3x+6 &amp;= 3x+6 &amp;&amp; \\text{distribute} \\\\\r\n3x+6-3x &amp;= 3x+6-3x &amp;&amp; \\text{subtract }3x \\\\\r\n6 &amp;= 6 &amp;&amp; \\text{always true}\r\n\\end{align}[\/latex]\r\n\r\nInfinitely many solutions.<\/li>\r\n \t<li data-start=\"902\" data-end=\"1000\">\r\n<p data-start=\"905\" data-end=\"1000\">[latex]2x-5 = 2x+1[\/latex]<\/p>\r\n[latex]\\begin{align}\r\n2x-5 &amp;= 2x+1 \\\\\r\n2x-5-2x &amp;= 2x+1-2x &amp;&amp; \\text{subtract }2x \\\\\r\n-5 &amp;= 1 &amp;&amp; \\text{false}\r\n\\end{align}[\/latex]\r\n\r\nNo solution.<\/li>\r\n \t<li data-start=\"1002\" data-end=\"1111\">\r\n<p data-start=\"1005\" data-end=\"1111\">[latex]4x+1 = 5x-3[\/latex]<\/p>\r\n[latex]\\begin{align} 4x+1 &amp;= 5x-3 \\\\\r\n4x-5x+1 &amp;= 5x -3 -5x &amp;&amp; \\text{subtract }5x \\\\\r\n-x+1 &amp;= -3 \\\\\r\n-x +1 - 1&amp;= -3 - 1 &amp;&amp; \\text{subtract }1 \\\\\r\n- x &amp;= - 4 \\\\\r\nx &amp;=\u00a0 4 &amp;&amp; \\text{divide by }-1\r\n\\end{align}[\/latex]\r\n\r\nOne solution: [latex]x=4[\/latex]<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]312214[\/ohm_question]<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">Combine like terms on each side of the equation before deciding the number of solutions.<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Recognize when a linear equation has no solution or infinite solutions<\/span><\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>no solution \/ infinite solutions<\/h3>\n<p data-start=\"328\" data-end=\"434\">When solving a linear equation, sometimes the variable cancels out. What remains tells you the number of solutions:<\/p>\n<ul data-start=\"436\" data-end=\"691\">\n<li data-start=\"436\" data-end=\"537\">\n<p data-start=\"438\" data-end=\"537\">If you reach a true statement like [latex]0=0[\/latex], there are <strong data-start=\"507\" data-end=\"536\">infinitely many solutions<\/strong>.<\/p>\n<\/li>\n<li data-start=\"538\" data-end=\"625\">\n<p data-start=\"540\" data-end=\"625\">If you reach a false statement like [latex]0=5[\/latex], there is <strong data-start=\"609\" data-end=\"624\">no solution<\/strong>.<\/p>\n<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p data-start=\"705\" data-end=\"749\">Decide how many solutions each equation has.<\/p>\n<ol data-start=\"751\" data-end=\"1111\">\n<li data-start=\"751\" data-end=\"900\">\n<p data-start=\"754\" data-end=\"900\">[latex]3x+6 = 3(x+2)[\/latex]<\/p>\n<p>[latex]\\begin{align}  3x+6 &= 3(x+2) \\\\  3x+6 &= 3x+6 && \\text{distribute} \\\\  3x+6-3x &= 3x+6-3x && \\text{subtract }3x \\\\  6 &= 6 && \\text{always true}  \\end{align}[\/latex]<\/p>\n<p>Infinitely many solutions.<\/li>\n<li data-start=\"902\" data-end=\"1000\">\n<p data-start=\"905\" data-end=\"1000\">[latex]2x-5 = 2x+1[\/latex]<\/p>\n<p>[latex]\\begin{align}  2x-5 &= 2x+1 \\\\  2x-5-2x &= 2x+1-2x && \\text{subtract }2x \\\\  -5 &= 1 && \\text{false}  \\end{align}[\/latex]<\/p>\n<p>No solution.<\/li>\n<li data-start=\"1002\" data-end=\"1111\">\n<p data-start=\"1005\" data-end=\"1111\">[latex]4x+1 = 5x-3[\/latex]<\/p>\n<p>[latex]\\begin{align} 4x+1 &= 5x-3 \\\\  4x-5x+1 &= 5x -3 -5x && \\text{subtract }5x \\\\  -x+1 &= -3 \\\\  -x +1 - 1&= -3 - 1 && \\text{subtract }1 \\\\  - x &= - 4 \\\\  x &=\u00a0 4 && \\text{divide by }-1  \\end{align}[\/latex]<\/p>\n<p>One solution: [latex]x=4[\/latex]<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm312214\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=312214&theme=lumen&iframe_resize_id=ohm312214&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">Combine like terms on each side of the equation before deciding the number of solutions.<\/section>\n","protected":false},"author":67,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":131,"module-header":"background_you_need","content_attributions":[],"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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2644"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":9,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2644\/revisions"}],"predecessor-version":[{"id":4033,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2644\/revisions\/4033"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/131"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2644\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2644"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2644"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2644"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2644"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}