{"id":10894,"date":"2022-06-06T19:20:17","date_gmt":"2022-06-06T13:50:17","guid":{"rendered":"https:\/\/mathemerize.com\/?p=10894"},"modified":"2022-06-06T19:20:20","modified_gmt":"2022-06-06T13:50:20","slug":"on-comparing-the-ratios-a_1over-a_2-b_1over-b_2-and-c_1over-c_2-find-out-whether-the-following-pair-of-linear-equations-are-consistent-or-inconsistent","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/on-comparing-the-ratios-a_1over-a_2-b_1over-b_2-and-c_1over-c_2-find-out-whether-the-following-pair-of-linear-equations-are-consistent-or-inconsistent\/","title":{"rendered":"On comparing the ratios \\(a_1\\over a_2\\), \\(b_1\\over b_2\\) and \\(c_1\\over c_2\\), find out whether the following pair of linear equations are consistent, or inconsistent."},"content":{"rendered":"

Question<\/strong> : On comparing the ratios \\(a_1\\over a_2\\), \\(b_1\\over b_2\\) and \\(c_1\\over c_2\\), find out whether the following pair of linear equations are consistent, or inconsistent.<\/p>\n

(i)<\/strong>\u00a0 3x + 2y = 5; 2x – 3y = 7<\/p>\n

(ii)<\/strong>\u00a0 2x – 3y = 8; 4x – 6y = 9<\/p>\n

(iii)<\/strong>\u00a0 \\(3\\over 2\\)x + \\(5\\over 3\\)y = 7;\u00a0 9x – 10y = 14<\/p>\n

(iv)<\/strong>\u00a0 5x – 3y = 11;\u00a0 -10x + 6y = -22<\/p>\n

(v)<\/strong>\u00a0 \\(4\\over 3\\)x + 2y = 8;\u00a0 2x + 3y = 12<\/p>\n

Solution<\/strong> :<\/p>\n

(i)<\/strong>\u00a0 Rewrite the given equations as:<\/p>\n

3x + 2y – 5 = 0;\u00a0 2x – 3y – 7 = 0<\/p>\n

\\(a_1\\) = 3, \\(b_1\\) = 2, \\(c_1\\) = 5<\/p>\n

\\(a_2\\) = 2, \\(b_2\\) = -3, \\(c_2\\) = -7<\/p>\n

\\(a_1\\over a_2\\) = \\(3\\over 2\\),\u00a0 \\(b_1\\over b_2\\) = \\(2\\over -3\\)<\/p>\n

Thus,\u00a0 \u00a0\\(3\\over 2\\)\u00a0 \\(\\ne\\)\u00a0 \\(2\\over -3\\),\u00a0 i.e.\u00a0 \\(a_1\\over a_2\\)\u00a0 \\(\\ne\\)\u00a0 \\(b_1\\over b_2\\)<\/p>\n

Hence, the pair of linear equations is consistent<\/strong>.<\/p>\n

(ii)<\/strong>\u00a0 Rewrite the given equations as:<\/p>\n

2x – 3y – 8 = 0;\u00a0 4x – 6y – 9 = 0<\/p>\n

\\(a_1\\) = 2, \\(b_1\\) = -3, \\(c_1\\) = -8<\/p>\n

\\(a_2\\) = 4, \\(b_2\\) = -6, \\(c_2\\) = -9<\/p>\n

\\(a_1\\over a_2\\) = \\(2\\over 4\\) = \\(1\\over 2\\),\u00a0 \\(b_1\\over b_2\\) = \\(-3\\over -6\\) = \\(1\\over 2\\), \\(c_1\\over c_2\\) = \\(-8\\over -9\\) = \\(8\\over 9\\)<\/p>\n

Thus,\u00a0 \u00a0\\(1\\over 2\\)\u00a0 = \\(1\\over 2\\) \\(\\ne\\)\u00a0 \\(8\\over 9\\),\u00a0 i.e.\u00a0 \\(a_1\\over a_2\\)\u00a0 = \\(b_1\\over b_2\\) \\(\\ne\\) \\(c_1\\over c_2\\)<\/p>\n

Hence, the pair of linear equations is inconsistent<\/strong>.<\/p>\n

(iii)<\/strong>\u00a0 Rewrite the given equations as:<\/p>\n

\\(3\\over 2\\)x + \\(5\\over 3\\)y – 7 = 0;\u00a0 9x – 10y – 14 = 0<\/p>\n

\\(a_1\\) = \\(3\\over 2\\), \\(b_1\\) = \\(5\\over 3\\), \\(c_1\\) = -7<\/p>\n

\\(a_2\\) = 9, \\(b_2\\) = -10, \\(c_2\\) = -14<\/p>\n

\\(a_1\\over a_2\\) = \\(1\\over 6\\),\u00a0 \\(b_1\\over b_2\\) = \\(-1\\over 6\\)<\/p>\n

Thus,\u00a0 \u00a0\\(1\\over 6\\)\u00a0 \\(\\ne\\)\u00a0 \\(-1\\over 6\\),\u00a0 i.e.\u00a0 \\(a_1\\over a_2\\)\u00a0 \\(\\ne\\)\u00a0 \\(b_1\\over b_2\\)<\/p>\n

Hence, the pair of linear equations is consistent<\/strong>.<\/p>\n

(iv)<\/strong>\u00a0 Rewrite the given equations as:<\/p>\n

5x – 3y – 11 = 0;\u00a0 -10x + 6y + 22 = 0<\/p>\n

\\(a_1\\) = 5, \\(b_1\\) = -3, \\(c_1\\) = -11<\/p>\n

\\(a_2\\) = -10, \\(b_2\\) = 6, \\(c_2\\) = 22<\/p>\n

\\(a_1\\over a_2\\) = \\(-1\\over 2\\),\u00a0 \\(b_1\\over b_2\\) = \\(-1\\over 2\\), \\(c_1\\over c_2\\) = \\(-1\\over 2\\)<\/p>\n

Thus,\u00a0 \u00a0\\(-1\\over 2\\) = \\(-1\\over 2\\) = \\(-1\\over 2\\),\u00a0 i.e.\u00a0 \\(a_1\\over a_2\\) = \\(b_1\\over b_2\\) = \\(c_1\\over c_2\\)<\/p>\n

Hence, the pair of linear equations is consistent (or dependent)<\/strong>.<\/p>\n

(v)<\/strong>\u00a0 Rewrite the given equations as:<\/p>\n

\\(4\\over 3\\)x + 2y – 8 = 0;\u00a0 2x + 3y – 12 = 0<\/p>\n

\\(a_1\\) = \\(4\\over 3\\), \\(b_1\\) = 2, \\(c_1\\) = -8<\/p>\n

\\(a_2\\) = 2, \\(b_2\\) = 3, \\(c_2\\) = -12<\/p>\n

\\(a_1\\over a_2\\) = \\(2\\over 3\\),\u00a0 \\(b_1\\over b_2\\) = \\(2\\over 3\\), \\(c_1\\over c_2\\) = \\(2\\over 3\\)<\/p>\n

Thus,\u00a0 \u00a0\\(2\\over 3\\) = \\(2\\over 3\\) = \\(2\\over 3\\),\u00a0 i.e.\u00a0 \\(a_1\\over a_2\\) = \\(b_1\\over b_2\\) = \\(c_1\\over c_2\\)<\/p>\n

Hence, the pair of linear equations is consistent (or dependent)<\/strong>.<\/p>\n","protected":false},"excerpt":{"rendered":"

Question : On comparing the ratios \\(a_1\\over a_2\\), \\(b_1\\over b_2\\) and \\(c_1\\over c_2\\), find out whether the following pair of linear equations are consistent, or inconsistent. (i)\u00a0 3x + 2y = 5; 2x – 3y = 7 (ii)\u00a0 2x – 3y = 8; 4x – 6y = 9 (iii)\u00a0 \\(3\\over 2\\)x + \\(5\\over 3\\)y = …<\/p>\n

On comparing the ratios \\(a_1\\over a_2\\), \\(b_1\\over b_2\\) and \\(c_1\\over c_2\\), find out whether the following pair of linear equations are consistent, or inconsistent.<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","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":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[911,43],"tags":[],"yoast_head":"\nOn comparing the ratios \\(a_1\\over a_2\\), \\(b_1\\over b_2\\) and \\(c_1\\over c_2\\), find out whether the following pair of linear equations are consistent, or inconsistent. - Mathemerize<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/on-comparing-the-ratios-a_1over-a_2-b_1over-b_2-and-c_1over-c_2-find-out-whether-the-following-pair-of-linear-equations-are-consistent-or-inconsistent\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"On comparing the ratios \\(a_1\\over a_2\\), \\(b_1\\over b_2\\) and \\(c_1\\over c_2\\), find out whether the following pair of linear equations are consistent, or inconsistent. - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"Question : On comparing the ratios (a_1over a_2), (b_1over b_2) and (c_1over c_2), find out whether the following pair of linear equations are consistent, or inconsistent. 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