{"id":676938,"date":"2023-09-02T19:44:29","date_gmt":"2023-09-02T14:14:29","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/question\/for-what-value-of-k-the-system-of-the-following-linear-equations-has-infinitely-many-solutionskx4yk-4-and-16xkyk\/"},"modified":"2025-06-20T14:45:52","modified_gmt":"2025-06-20T09:15:52","slug":"for-what-value-of-k-the-system-of-the-following-linear-equations-has-infinitely-many-solutionskx4yk-4-and-16xkyk","status":"publish","type":"questions","link":"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/for-what-value-of-k-the-system-of-the-following-linear-equat\/","title":{"rendered":"For what value of k the system of the following linear equations has infinitely many solutions?kx+4y=k-4 and 16x+ky=k."},"content":{"rendered":"","protected":false},"author":1,"template":"","meta":{"_yoast_wpseo_focuskw":"","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"For what value of k the system of the following linear equations has infinitely many solutions?kx+4y=k-4 and 16x+ky=k.","custom_permalink":"question\/mathematics\/for-what-value-of-k-the-system-of-the-following-linear-equat\/"},"categories":[],"acf":{"question":"

For what value of <\/span>k<\/mi><\/math> the system of the following linear equations has infinitely many solutions?<\/span><\/p>

kx<\/mi>+<\/mo>4<\/mn>y<\/mi>=<\/mo>k<\/mi>-<\/mo>4<\/mn><\/math> and <\/span>16<\/mn>x<\/mi>+<\/mo>ky<\/mi>=<\/mo>k<\/mi><\/math>.<\/span><\/p>

<\/p>

<\/p><\/spanstyle=\"font-family:'timesnewroman';-aw-import:ignore\"><\/pstyle=\"margin-top:0pt;margin-bottom:0pt;line-height:normal;font-size:12pt\"><\/spanstyle=\"font-family:'timesnewroman';-aw-import:ignore\"><\/pstyle=\"margin-top:0pt;margin-bottom:0pt;line-height:normal;font-size:12pt\">","options":[{"option":"0<\/mn><\/math> <\/span>","correct":false},{"option":"8<\/mn><\/math> <\/span>","correct":true},{"option":"-<\/mo>8<\/mn><\/math> <\/span>","correct":false},{"option":"4<\/mn><\/math> <\/span> <\/span>","correct":false}],"solution":"Given,<\/span>
kx<\/mi>+<\/mo>4<\/mn>y<\/mi>=<\/mo>k<\/mi>-<\/mo>4<\/mn><\/math> ,<\/span>
16<\/mn>x<\/mi>+<\/mo>ky<\/mi>=<\/mo>k<\/mi><\/math> .<\/span>
Here, <\/span>a<\/mi><\/mrow>1<\/mn><\/mrow><\/msub>=<\/mo>k<\/mi>,<\/mo> b<\/mi><\/mrow>1<\/mn><\/mrow><\/msub>=<\/mo>4<\/mn>,<\/mo> c<\/mi><\/mrow>1<\/mn><\/mrow><\/msub>=<\/mo>-<\/mo>(<\/mo>k<\/mi>-<\/mo>4<\/mn>)<\/mo><\/math>.<\/span>
And <\/span>a<\/mi><\/mrow>2<\/mn><\/mrow><\/msub>=<\/mo>16<\/mn>,<\/mo> <\/mi>b<\/mi><\/mrow>2<\/mn><\/mrow><\/msub>=<\/mo>k<\/mi>,<\/mo> <\/mi>c<\/mi><\/mrow>2<\/mn><\/mrow><\/msub>=<\/mo>-<\/mo>k<\/mi><\/math>.<\/span>
The given equations are consistent if,<\/span>
a<\/mi><\/mrow>1<\/mn><\/mrow><\/msub><\/mrow>a<\/mi><\/mrow>2<\/mn><\/mrow><\/msub><\/mrow><\/mfrac>=<\/mo>b<\/mi><\/mrow>1<\/mn><\/mrow><\/msub><\/mrow>b<\/mi><\/mrow>2<\/mn><\/mrow><\/msub><\/mrow><\/mfrac>=<\/mo>c<\/mi><\/mrow>1<\/mn><\/mrow><\/msub><\/mrow>c<\/mi><\/mrow>2<\/mn><\/mrow><\/msub><\/mrow><\/mfrac><\/math> <\/span>\u21d2 <\/span>k<\/mi><\/mrow>16<\/mn><\/mrow><\/mfrac>=<\/mo>4<\/mn><\/mrow>k<\/mi><\/mrow><\/mfrac>=<\/mo>k<\/mi>-<\/mo>4<\/mn><\/mrow>k<\/mi><\/mrow><\/mfrac><\/math> <\/span>\u21d2 <\/span>k<\/mi><\/mrow>16<\/mn><\/mrow><\/mfrac>=<\/mo>4<\/mn><\/mrow>k<\/mi><\/mrow><\/mfrac><\/math>
\u21d2 <\/span>k<\/mi><\/mrow>2<\/mn><\/mrow><\/msup>=<\/mo>64<\/mn><\/math>
\u21d2 <\/span>k<\/mi>=<\/mo>\u00b1<\/mo>8<\/mn><\/math> <\/span>Also, <\/span>4<\/mn><\/mrow>k<\/mi><\/mrow><\/mfrac>=<\/mo>k<\/mi>-<\/mo>4<\/mn><\/mrow>k<\/mi><\/mrow><\/mfrac><\/math>.<\/span>
\u21d2 <\/span>4<\/mn>k<\/mi>=<\/mo>k<\/mi><\/mrow>2<\/mn><\/mrow><\/msup>-<\/mo>4<\/mn>k<\/mi><\/math> <\/span>\u21d2 <\/span>k<\/mi><\/mrow>2<\/mn><\/mrow><\/msup>-<\/mo>8<\/mn>k<\/mi>=<\/mo>0<\/mn><\/math>
\u21d2 <\/span>k<\/mi>(<\/mo>k<\/mi>-<\/mo>8<\/mn>)<\/mo>=<\/mo>0<\/mn><\/math> <\/span>\u21d2 <\/span>k<\/mi>=<\/mo>0<\/mn> <\/mi><\/math>or <\/span>k<\/mi>=<\/mo>8<\/mn><\/math>
But <\/span>k<\/mi>=<\/mo>0<\/mn><\/math> is not possible. <\/span>
Therefore, <\/span>k<\/mi>=<\/mo> <\/mi>8<\/mn><\/math> is the correct value for infinitely many solutions. <\/span>
Hence, the correct option is 2.<\/span>
 <\/span>","subject":"Mathematics"},"yoast_head":"\nFor what value of k the system of the following linear equations has infinitely many solutions?kx+4y=k-4 and 16x+ky=k. - Infinity Learn by Sri Chaitanya<\/title>\n<meta name=\"description\" content=\"For what value of k the system of the following linear equations has infinitely many solutions?kx+4y=k-4 and 16x+ky=k.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/for-what-value-of-k-the-system-of-the-following-linear-equat\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"For what value of k the system of the following linear equations has infinitely many solutions?kx+4y=k-4 and 16x+ky=k. - Infinity Learn by Sri Chaitanya\" \/>\n<meta property=\"og:description\" content=\"For what value of k the system of the following linear equations has infinitely many solutions?kx+4y=k-4 and 16x+ky=k.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/for-what-value-of-k-the-system-of-the-following-linear-equat\/\" \/>\n<meta property=\"og:site_name\" content=\"Infinity Learn by Sri Chaitanya\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/InfinityLearn.SriChaitanya\/\" \/>\n<meta property=\"article:modified_time\" content=\"2025-06-20T09:15:52+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2025\/04\/infinitylearn.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1920\" \/>\n\t<meta property=\"og:image:height\" content=\"1008\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:site\" content=\"@InfinityLearn_\" \/>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"For what value of k the system of the following linear equations has infinitely many solutions?kx+4y=k-4 and 16x+ky=k. - Infinity Learn by Sri Chaitanya","description":"For what value of k the system of the following linear equations has infinitely many solutions?kx+4y=k-4 and 16x+ky=k.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/for-what-value-of-k-the-system-of-the-following-linear-equat\/","og_locale":"en_US","og_type":"article","og_title":"For what value of k the system of the following linear equations has infinitely many solutions?kx+4y=k-4 and 16x+ky=k. - Infinity Learn by Sri Chaitanya","og_description":"For what value of k the system of the following linear equations has infinitely many solutions?kx+4y=k-4 and 16x+ky=k.","og_url":"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/for-what-value-of-k-the-system-of-the-following-linear-equat\/","og_site_name":"Infinity Learn by Sri Chaitanya","article_publisher":"https:\/\/www.facebook.com\/InfinityLearn.SriChaitanya\/","article_modified_time":"2025-06-20T09:15:52+00:00","og_image":[{"width":1920,"height":1008,"url":"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2025\/04\/infinitylearn.jpg","type":"image\/jpeg"}],"twitter_card":"summary_large_image","twitter_site":"@InfinityLearn_","schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Organization","@id":"https:\/\/infinitylearn.com\/surge\/#organization","name":"Infinity Learn","url":"https:\/\/infinitylearn.com\/surge\/","sameAs":["https:\/\/www.facebook.com\/InfinityLearn.SriChaitanya\/","https:\/\/www.instagram.com\/infinitylearn_by_srichaitanya\/","https:\/\/www.linkedin.com\/company\/infinity-learn-by-sri-chaitanya\/","https:\/\/www.youtube.com\/c\/InfinityLearnEdu","https:\/\/twitter.com\/InfinityLearn_"],"logo":{"@type":"ImageObject","@id":"https:\/\/infinitylearn.com\/surge\/#logo","inLanguage":"en-US","url":"","contentUrl":"","caption":"Infinity Learn"},"image":{"@id":"https:\/\/infinitylearn.com\/surge\/#logo"}},{"@type":"WebSite","@id":"https:\/\/infinitylearn.com\/surge\/#website","url":"https:\/\/infinitylearn.com\/surge\/","name":"Infinity Learn by Sri Chaitanya","description":"Surge","publisher":{"@id":"https:\/\/infinitylearn.com\/surge\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/infinitylearn.com\/surge\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/for-what-value-of-k-the-system-of-the-following-linear-equat\/#webpage","url":"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/for-what-value-of-k-the-system-of-the-following-linear-equat\/","name":"For what value of k the system of the following linear equations has infinitely many solutions?kx+4y=k-4 and 16x+ky=k. - Infinity Learn by Sri Chaitanya","isPartOf":{"@id":"https:\/\/infinitylearn.com\/surge\/#website"},"datePublished":"2023-09-02T14:14:29+00:00","dateModified":"2025-06-20T09:15:52+00:00","description":"For what value of k the system of the following linear equations has infinitely many solutions?kx+4y=k-4 and 16x+ky=k.","breadcrumb":{"@id":"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/for-what-value-of-k-the-system-of-the-following-linear-equat\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/infinitylearn.com\/surge\/question\/mathematics\/for-what-value-of-k-the-system-of-the-following-linear-equat\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/for-what-value-of-k-the-system-of-the-following-linear-equat\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/infinitylearn.com\/surge\/"},{"@type":"ListItem","position":2,"name":"For what value of k the system of the following linear equations has infinitely many solutions?kx+4y=k-4 and 16x+ky=k."}]}]}},"_links":{"self":[{"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/questions\/676938"}],"collection":[{"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/questions"}],"about":[{"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/types\/questions"}],"author":[{"embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/users\/1"}],"wp:attachment":[{"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/media?parent=676938"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/categories?post=676938"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}