{"id":10910,"date":"2022-06-07T15:41:50","date_gmt":"2022-06-07T10:11:50","guid":{"rendered":"https:\/\/mathemerize.com\/?p=10910"},"modified":"2022-06-07T15:48:27","modified_gmt":"2022-06-07T10:18:27","slug":"which-of-the-following-pairs-of-linear-equations-are-consistent-obtain-the-solution-in-such-cases-graphically","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/which-of-the-following-pairs-of-linear-equations-are-consistent-obtain-the-solution-in-such-cases-graphically\/","title":{"rendered":"Which of the following pairs of linear equations are consistent obtain the solution in such cases graphically."},"content":{"rendered":"\n
Question<\/strong> : Which of the following pairs of linear equations are consistent obtain the solution in such cases graphically.<\/p>\n\n\n\n (i)<\/strong> x + y = 5, 2x + 2y = 10<\/p>\n\n\n\n (ii) <\/strong> x – y = 8, 3x – 3y = 10<\/p>\n\n\n\n (iii)<\/strong> 2x + y – 6 = 0, 4x – 2y – 4 = 0<\/p>\n\n\n\n (iv)<\/strong> 2x – 2y – 2 = 0, 4x – 4y – 5 = 0<\/p>\n\n\n\n Solution<\/strong> : <\/p>\n\n\n\n (i)<\/strong> We have the given equations,<\/p>\n\n\n\n x + y = 5 \\(\\implies\\) y = 5 – x<\/p>\n\n\n\n If x = 0, y = 5<\/p>\n\n\n\n If x = 5, y = 0<\/p>\n\n\n\n and 2x + 2y = 10 \\(\\implies\\) y = \\(10 – 2x\\over 2\\)<\/p>\n\n\n\n If x = 0, y = 5<\/p>\n\n\n\n If x = 2, y = 3<\/p>\n\n\n\n If x = 5, y = 0<\/p>\n\n\n\n Now, plot these points in the table on graph as shown in figure below. By plotting A and B points we get the line AB and by plotting C, D and E points we get the line CD.<\/p>\n\n\n\n We see that both the lines in the graph are coincident. Therefore both equations have infinitely many solutions.<\/strong><\/p>\n\n\n\n Hence, the pair of linear equations is consistent.<\/strong><\/p>\n\n\n\n (ii)<\/strong> We have the given equations,<\/p>\n\n\n\n x – y = 8 \\(\\implies\\) y = x – 8<\/p>\n\n\n\n If x = 0, y = -8<\/p>\n\n\n\n If x = 8, y = 0<\/p>\n\n\n\n and 3x – 3y = 16 \\(\\implies\\) y = \\(3x – 16\\over 3\\)<\/p>\n\n\n\n If x = 0, y = \\(-16\\over 3\\)<\/p>\n\n\n\n If x = 2, y = \\(-10\\over 3\\)<\/p>\n\n\n\n Now, plot these points in the table on graph as shown in figure below. By plotting A and B points we get the line AB and by plotting C and D points we get the line CD.<\/p>\n\n\n\n We see that both the lines are parallel. Therefore both equations have has no solution.<\/strong><\/p>\n\n\n\n Hence, the pair of linear equations is inconsistent.<\/strong><\/p>\n\n\n\n (iii)<\/strong> We have the given equations,<\/p>\n\n\n\n 2x + y – 6 = 0 \\(\\implies\\) y = 6 – 2x<\/p>\n\n\n\n If x = 0, y = 6<\/p>\n\n\n\n If x = 3, y = 0<\/p>\n\n\n\n and 4x – 2y – 4 = 0 \\(\\implies\\) y = 2x – 2<\/p>\n\n\n\n If x = 0, y = 1<\/p>\n\n\n\n If x = -2, y = 0<\/p>\n\n\n\n Now, plot these points in the table on graph as shown in figure below. By plotting A and B points we get the straight line AB and by plotting C, D points we get the line CD.<\/p>\n\n\n\n The lines AB and CD intersect at E.<\/p>\n\n\n\n We see that both the lines in the graph have a common point E. Therefore the given equations is consistent and this point E gives us the solution of both the lines.<\/strong><\/p>\n\n\n\n Hence, the pair of linear equations is consistent.<\/strong><\/p>\n\n\n\n (iv)<\/strong> We have the given equations,<\/p>\n\n\n\n 2x – 2y – 2 = 0 \\(\\implies\\) y = x – 1<\/p>\n\n\n\n If x = 0, y = -1<\/p>\n\n\n\n If x = 1, y = 0<\/p>\n\n\n\n and 4x – 4y – 5 = 0 \\(\\implies\\) y = \\(x – {5\\over 4}\\)<\/p>\n\n\n\n If x = 0, y = \\(-5\\over 4\\)<\/p>\n\n\n\n If x = \\(5\\over 4\\), y = 0<\/p>\n\n\n\n Now, plot these points in the table on graph as shown in figure below. By plotting A and B points we get the line AB and by plotting C and D points we get the line CD.<\/p>\n\n\n\n We see that both the lines are parallel. Therefore both equations have has no solution.<\/strong><\/p>\n\n\n\n Hence, the pair of linear equations is inconsistent.<\/strong><\/p>\n\n\n\nx<\/td> 0<\/td> 5<\/td><\/tr> y = 5 – x<\/td> 5<\/td> 0<\/td><\/tr> Points<\/td> A<\/td> B<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n x<\/td> 0<\/td> 2<\/td> 5<\/td><\/tr> y = \\(10 – 2x\\over 2\\)<\/td> 5<\/td> 3<\/td> 0<\/td><\/tr> Points<\/td> C<\/td> D<\/td> E<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n x<\/td> 0<\/td> 8<\/td><\/tr> y = x – 8<\/td> -8<\/td> 0<\/td><\/tr> Points<\/td> A<\/td> B<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n x<\/td> 0<\/td> 2<\/td><\/tr> y = \\(3x – 16\\over 3\\)<\/td> \\(-16\\over 3\\)<\/td> \\(-10\\over 3\\)<\/td><\/tr> Points<\/td> C<\/td> D<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n x<\/td> 0<\/td> 3<\/td><\/tr> y<\/td> 6<\/td> 0<\/td><\/tr> Points<\/td> A<\/td> B<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n x<\/td> 0<\/td> 1<\/td><\/tr> y<\/td> -2<\/td> 0<\/td><\/tr> Point<\/td> D<\/td> E<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n x<\/td> 0<\/td> 1<\/td><\/tr> y<\/td> -1<\/td> 0<\/td><\/tr> Points<\/td> A<\/td> B<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n x<\/td> 0<\/td> \\(-5\\over 4\\)<\/td><\/tr> y<\/td> \\(5\\over 4\\)<\/td> 0<\/td><\/tr> Points<\/td> C<\/td> D<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n