{"id":191071,"date":"2025-02-14T06:07:09","date_gmt":"2025-02-14T06:07:09","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=191071"},"modified":"2025-02-14T06:07:11","modified_gmt":"2025-02-14T06:07:11","slug":"solve-each-equation","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/14\/solve-each-equation\/","title":{"rendered":"Solve each equation"},"content":{"rendered":"\n<p>Solve each equation. Remember to check for extraneous solutions.<br>1) sqrt(81x) &#8211; 5 = 4<br>2) sqrt(2a &#8211; 17) = sqrt(10 &#8211; a)<br>3) &#8211; 4 = &#8211; 9 + sqrt(3n + 7)<br>4) 6 = 1 + sqrt(5x)<br>5) 49 = 7sqrt(- 1 &#8211; 10n)<br>6) sqrt(19a &#8211; 2) + 5 = 11<br>7) sqrt(15 &#8211; r) = sqrt(2r &#8211; 15)<br>8) sqrt(2x + 7) = sqrt(3x + 6)<br>9) sqrt(3a &#8211; 13) = sqrt(2a &#8211; 8)<br>10) 3 = sqrt(3v)<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>Let\u2019s solve each equation step by step and then check for any extraneous solutions that might arise.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">1) ( \\sqrt{81x} &#8211; 5 = 4 )<\/h3>\n\n\n\n<p><strong>Step 1:<\/strong> Add 5 to both sides:<br>[<br>\\sqrt{81x} = 9<br>]<\/p>\n\n\n\n<p><strong>Step 2:<\/strong> Square both sides to eliminate the square root:<br>[<br>81x = 81<br>]<\/p>\n\n\n\n<p><strong>Step 3:<\/strong> Divide by 81:<br>[<br>x = 1<br>]<\/p>\n\n\n\n<p><strong>Check:<\/strong> Substitute ( x = 1 ) back into the original equation:<br>[<br>\\sqrt{81(1)} &#8211; 5 = 4 \\quad \\Rightarrow \\quad 9 &#8211; 5 = 4<br>]<br>Since the original equation holds true, the solution is correct: ( x = 1 ).<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">2) ( \\sqrt{2a &#8211; 17} = \\sqrt{10 &#8211; a} )<\/h3>\n\n\n\n<p><strong>Step 1:<\/strong> Square both sides:<br>[<br>2a &#8211; 17 = 10 &#8211; a<br>]<\/p>\n\n\n\n<p><strong>Step 2:<\/strong> Add ( a ) to both sides:<br>[<br>3a &#8211; 17 = 10<br>]<\/p>\n\n\n\n<p><strong>Step 3:<\/strong> Add 17 to both sides:<br>[<br>3a = 27<br>]<\/p>\n\n\n\n<p><strong>Step 4:<\/strong> Divide by 3:<br>[<br>a = 9<br>]<\/p>\n\n\n\n<p><strong>Check:<\/strong> Substitute ( a = 9 ) back into the original equation:<br>[<br>\\sqrt{2(9) &#8211; 17} = \\sqrt{10 &#8211; 9} \\quad \\Rightarrow \\quad \\sqrt{18 &#8211; 17} = \\sqrt{1} \\quad \\Rightarrow \\quad 1 = 1<br>]<br>Since this is true, the solution is correct: ( a = 9 ).<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">3) ( -4 = -9 + \\sqrt{3n + 7} )<\/h3>\n\n\n\n<p><strong>Step 1:<\/strong> Add 9 to both sides:<br>[<br>5 = \\sqrt{3n + 7}<br>]<\/p>\n\n\n\n<p><strong>Step 2:<\/strong> Square both sides:<br>[<br>25 = 3n + 7<br>]<\/p>\n\n\n\n<p><strong>Step 3:<\/strong> Subtract 7 from both sides:<br>[<br>18 = 3n<br>]<\/p>\n\n\n\n<p><strong>Step 4:<\/strong> Divide by 3:<br>[<br>n = 6<br>]<\/p>\n\n\n\n<p><strong>Check:<\/strong> Substitute ( n = 6 ) back into the original equation:<br>[<br>-4 = -9 + \\sqrt{3(6) + 7} \\quad \\Rightarrow \\quad -4 = -9 + \\sqrt{18 + 7} \\quad \\Rightarrow \\quad -4 = -9 + \\sqrt{25} \\quad \\Rightarrow \\quad -4 = -9 + 5<br>]<br>Since this is true, the solution is correct: ( n = 6 ).<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">4) ( 6 = 1 + \\sqrt{5x} )<\/h3>\n\n\n\n<p><strong>Step 1:<\/strong> Subtract 1 from both sides:<br>[<br>5 = \\sqrt{5x}<br>]<\/p>\n\n\n\n<p><strong>Step 2:<\/strong> Square both sides:<br>[<br>25 = 5x<br>]<\/p>\n\n\n\n<p><strong>Step 3:<\/strong> Divide by 5:<br>[<br>x = 5<br>]<\/p>\n\n\n\n<p><strong>Check:<\/strong> Substitute ( x = 5 ) back into the original equation:<br>[<br>6 = 1 + \\sqrt{5(5)} \\quad \\Rightarrow \\quad 6 = 1 + \\sqrt{25} \\quad \\Rightarrow \\quad 6 = 1 + 5<br>]<br>Since this is true, the solution is correct: ( x = 5 ).<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">5) ( 49 = 7\\sqrt{-1 &#8211; 10n} )<\/h3>\n\n\n\n<p><strong>Step 1:<\/strong> Divide by 7:<br>[<br>7 = \\sqrt{-1 &#8211; 10n}<br>]<\/p>\n\n\n\n<p><strong>Step 2:<\/strong> Square both sides:<br>[<br>49 = -1 &#8211; 10n<br>]<\/p>\n\n\n\n<p><strong>Step 3:<\/strong> Add 1 to both sides:<br>[<br>50 = -10n<br>]<\/p>\n\n\n\n<p><strong>Step 4:<\/strong> Divide by -10:<br>[<br>n = -5<br>]<\/p>\n\n\n\n<p><strong>Check:<\/strong> Substitute ( n = -5 ) back into the original equation:<br>[<br>49 = 7\\sqrt{-1 &#8211; 10(-5)} \\quad \\Rightarrow \\quad 49 = 7\\sqrt{-1 + 50} \\quad \\Rightarrow \\quad 49 = 7\\sqrt{49} \\quad \\Rightarrow \\quad 49 = 7 \\times 7<br>]<br>Since this is true, the solution is correct: ( n = -5 ).<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">6) ( \\sqrt{19a &#8211; 2} + 5 = 11 )<\/h3>\n\n\n\n<p><strong>Step 1:<\/strong> Subtract 5 from both sides:<br>[<br>\\sqrt{19a &#8211; 2} = 6<br>]<\/p>\n\n\n\n<p><strong>Step 2:<\/strong> Square both sides:<br>[<br>19a &#8211; 2 = 36<br>]<\/p>\n\n\n\n<p><strong>Step 3:<\/strong> Add 2 to both sides:<br>[<br>19a = 38<br>]<\/p>\n\n\n\n<p><strong>Step 4:<\/strong> Divide by 19:<br>[<br>a = 2<br>]<\/p>\n\n\n\n<p><strong>Check:<\/strong> Substitute ( a = 2 ) back into the original equation:<br>[<br>\\sqrt{19(2) &#8211; 2} + 5 = 11 \\quad \\Rightarrow \\quad \\sqrt{38 &#8211; 2} + 5 = 11 \\quad \\Rightarrow \\quad \\sqrt{36} + 5 = 11 \\quad \\Rightarrow \\quad 6 + 5 = 11<br>]<br>Since this is true, the solution is correct: ( a = 2 ).<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">7) ( \\sqrt{15 &#8211; r} = \\sqrt{2r &#8211; 15} )<\/h3>\n\n\n\n<p><strong>Step 1:<\/strong> Square both sides:<br>[<br>15 &#8211; r = 2r &#8211; 15<br>]<\/p>\n\n\n\n<p><strong>Step 2:<\/strong> Add ( r ) to both sides:<br>[<br>15 = 3r &#8211; 15<br>]<\/p>\n\n\n\n<p><strong>Step 3:<\/strong> Add 15 to both sides:<br>[<br>30 = 3r<br>]<\/p>\n\n\n\n<p><strong>Step 4:<\/strong> Divide by 3:<br>[<br>r = 10<br>]<\/p>\n\n\n\n<p><strong>Check:<\/strong> Substitute ( r = 10 ) back into the original equation:<br>[<br>\\sqrt{15 &#8211; 10} = \\sqrt{2(10) &#8211; 15} \\quad \\Rightarrow \\quad \\sqrt{5} = \\sqrt{20 &#8211; 15} \\quad \\Rightarrow \\quad \\sqrt{5} = \\sqrt{5}<br>]<br>Since this is true, the solution is correct: ( r = 10 ).<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">8) ( \\sqrt{2x + 7} = \\sqrt{3x + 6} )<\/h3>\n\n\n\n<p><strong>Step 1:<\/strong> Square both sides:<br>[<br>2x + 7 = 3x + 6<br>]<\/p>\n\n\n\n<p><strong>Step 2:<\/strong> Subtract ( 2x ) from both sides:<br>[<br>7 = x + 6<br>]<\/p>\n\n\n\n<p><strong>Step 3:<\/strong> Subtract 6 from both sides:<br>[<br>x = 1<br>]<\/p>\n\n\n\n<p><strong>Check:<\/strong> Substitute ( x = 1 ) back into the original equation:<br>[<br>\\sqrt{2(1) + 7} = \\sqrt{3(1) + 6} \\quad \\Rightarrow \\quad \\sqrt{2 + 7} = \\sqrt{3 + 6} \\quad \\Rightarrow \\quad \\sqrt{9} = \\sqrt{9} \\quad \\Rightarrow \\quad 3 = 3<br>]<br>Since this is true, the solution is correct: ( x = 1 ).<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">9) ( \\sqrt{3a &#8211; 13} = \\sqrt{2a &#8211; 8} )<\/h3>\n\n\n\n<p><strong>Step 1:<\/strong> Square both sides:<br>[<br>3a &#8211; 13 = 2a &#8211; 8<br>]<\/p>\n\n\n\n<p><strong>Step 2:<\/strong> Subtract ( 2a ) from both sides:<br>[<br>a &#8211; 13 = -8<br>]<\/p>\n\n\n\n<p><strong>Step 3:<\/strong> Add 13 to both sides:<br>[<br>a = 5<br>]<\/p>\n\n\n\n<p><strong>Check:<\/strong> Substitute ( a = 5 ) back into the original equation:<br>[<br>\\sqrt{3(5) &#8211; 13} = \\sqrt{2(5) &#8211; 8} \\quad \\Rightarrow \\quad \\sqrt{15 &#8211; 13} = \\sqrt{10 &#8211; 8} \\quad \\Rightarrow \\quad \\sqrt{2} = \\sqrt{2}<br>]<br>Since this is true, the solution is correct: ( a = 5 ).<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">10) ( 3 = \\sqrt{3v} )<\/h3>\n\n\n\n<p><strong>Step 1:<\/strong> Square both sides:<br>[<br>9 = 3v<br>]<\/p>\n\n\n\n<p><strong>Step 2:<\/strong> Divide by 3:<br>[<br>v = 3<br>]<\/p>\n\n\n\n<p><strong>Check:<\/strong> Substitute ( v = 3 ) back into the original equation:<br>[<br>3 = \\sqrt{3(3)} \\quad \\Rightarrow \\quad 3 = \\sqrt{9} \\quad \\Rightarrow \\quad 3 = 3<br>]<br>Since this is true, the solution is correct: ( v = 3 ).<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Summary of solutions:<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>( x = 1 )<\/li>\n\n\n\n<li>( a = 9 )<\/li>\n\n\n\n<li>( n = 6 )<\/li>\n\n\n\n<li>( x = 5 )<\/li>\n\n\n\n<li>( n = -5 )<\/li>\n\n\n\n<li>( a = 2 )<\/li>\n\n\n\n<li>( r = 10 )<\/li>\n\n\n\n<li>( x = 1 )<\/li>\n\n\n\n<li>( a = 5 )<\/li>\n\n\n\n<li>( v = 3 )<\/li>\n<\/ol>\n\n\n\n<p>All solutions are valid with no extraneous solutions!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Solve each equation. Remember to check for extraneous solutions.1) sqrt(81x) &#8211; 5 = 42) sqrt(2a &#8211; 17) = sqrt(10 &#8211; a)3) &#8211; 4 = &#8211; 9 + sqrt(3n + 7)4) 6 = 1 + sqrt(5x)5) 49 = 7sqrt(- 1 &#8211; 10n)6) sqrt(19a &#8211; 2) + 5 = 117) sqrt(15 &#8211; r) = sqrt(2r &#8211; 15)8) [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-191071","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/191071","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=191071"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/191071\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=191071"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=191071"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=191071"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}