{"id":1002,"date":"2024-03-22T11:30:02","date_gmt":"2024-03-22T05:30:02","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=1002"},"modified":"2024-10-24T18:30:00","modified_gmt":"2024-10-24T12:30:00","slug":"solving-system-of-linear-equations","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/solving-system-of-linear-equations\/","title":{"rendered":"\u098f\u0995\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u099c\u09cb\u099f \u0993 \u09b8\u09ae\u09be\u09a7\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df (System of linear equation and it\u2019s Solution)"},"content":{"rendered":"

\"\u09b8\u09ae\u09c0\u0995\u09b0\u09a3<\/p>\n

\u098f\u0995\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u0995\u09be\u0995\u09c7 \u09ac\u09b2\u09c7? \u099c\u09cb\u099f (System of linear equations)<\/b><\/span><\/h3>\n

a_1x_1+a_2x_2+.....+a_nx_n=b<\/span> \u0995\u09c7 x_1,x_2,......,x_n<\/span><\/span> \u099a\u09b2\u0995\u09c7\u09b0 \u098f\u0995\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09ac\u09b2\u09be \u09b9\u09df, \u09af\u09c7\u0996\u09be\u09a8\u09c7 a_1,a_2,......,a_n<\/span><\/span>\u00a0\u09a7\u09cd\u09b0\u09c1\u09ac\u0995\u0964 \u098f\u0987\u09b0\u09c2\u09aa \u098f\u0995\u09be\u09a7\u09bf\u0995 \u098f\u0995\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u0995\u09c7 \u098f\u0995\u09a4\u09cd\u09b0\u09c7 \u098f\u0995\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u099c\u09cb\u099f<\/strong> \u09ac\u09b2\u09be \u09b9\u09df\u0964\u00a0<\/span><\/p>\n

\u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995\u09c7\u09b0<\/a><\/span> \u09b8\u09be\u09b9\u09be\u09af\u09cd\u09af\u09c7 \u09b8\u09ae\u09b8\u0982\u0996\u09cd\u09af\u0995 \u099a\u09b2\u0995 \u0993 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u098f\u0995\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u099c\u09cb\u099f\u09c7\u09b0 \u09b8\u09ae\u09be\u09a7\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u09be \u09af\u09be\u09df\u0964 \u09b8\u09ae\u09be\u09a7\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df\u09c7\u09b0 \u098f\u0987 \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf <\/span>\u09e7\u09ed\u09eb\u09e6 \u0996\u09cd\u09b0\u09bf\u09b7\u09cd\u099f\u09be\u09ac\u09cd\u09a6\u09c7 \u0997\u09cd\u09af\u09be\u09ac\u09bf\u09df\u09c7\u09b2 \u0995\u09cd\u09b0\u09c7\u09ae\u09be\u09b0<\/b> \u09aa\u09cd\u09b0\u09a4\u09bf\u09b7\u09cd\u09a0\u09be \u0995\u09b0\u09c7\u09a8 \u09ac\u09bf\u09a7\u09be\u09df \u098f\u0995\u09c7 \u0995\u09cd\u09b0\u09c7\u09ae\u09be\u09b0\u09c7\u09b0 \u09a8\u09bf\u09df\u09ae <\/span>(Cramer\u2019s Rule) <\/b>\u09ac\u09b2\u09be \u09b9\u09df\u0964\u00a0\u00a0\u00a0<\/span><\/p>\n

\u0995\u09cd\u09b0\u09c7\u09ae\u09be\u09b0\u09c7\u09b0 \u09a8\u09bf\u09df\u09ae\u09c7 \u098f\u0995\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u099c\u09cb\u099f \u0993 \u09b8\u09ae\u09be\u09a7\u09be\u09a8 (Solution of system of linear equations using cramer\u2019s rule)<\/b><\/span><\/h3>\n

\u09a6\u09c1\u0987 \u099a\u09b2\u0995\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u098f\u0995\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u099c\u09cb\u099f\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7-\u00a0\u00a0<\/b><\/h4>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u09aa\u09cd\u09b0\u09a6\u09a4\u09cd\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u099c\u09cb\u099f :<\/span> a_1x+b_1y=c_1......(i)\\\\\n\na_2x+b_2y=c_2......(ii)<\/span><\/span><\/p>\n

\u098f\u0996\u09be\u09a8\u09c7, <\/span>x\u00a0<\/span>\u0993 \u098f\u09b0 <\/span>y<\/span> \u09b8\u09b9\u0997\u0997\u09c1\u09b2\u09bf \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0997\u09a0\u09bf\u09a4 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995, D=\\begin{vmatrix}\n\na_1 & b_1\\\\\n\na_2 & b_2\n\n\\end{vmatrix} \\neq 0<\/span><\/span><\/p>\n

x\u00a0<\/span>\u098f\u09b0 \u09b8\u09b9\u0997\u09c7\u09b0 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09c7 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995 \u09aa\u09a6 \u09a8\u09bf\u09df\u09c7 \u0997\u09a0\u09bf\u09a4 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995, D_x=\\begin{vmatrix}\n\nc_1 & b_1\\\\\n\nc_2 & b_2\n\n\\end{vmatrix}<\/span><\/span><\/p>\n

\u098f\u09ac\u0982 <\/span>y\u00a0<\/span>\u098f\u09b0 \u09b8\u09b9\u0997\u09c7\u09b0 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09c7 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995 \u09aa\u09a6 \u09a8\u09bf\u09df\u09c7 \u0997\u09a0\u09bf\u09a4 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995, D_y=\\begin{vmatrix}\n\na_1 & c_1\\\\\n\na_2 & c_2\n\n\\end{vmatrix}<\/span><\/span><\/p>\n

\u09ac\u099c\u09cd\u09b0\u0997\u09c1\u09a3\u09a8 \u09b8\u09c2\u09a4\u09cd\u09b0\u09be\u09a8\u09c1\u09b8\u09be\u09b0\u09c7, \\frac{x}{-b_1c_2+b_2c_1}=\\frac{y}{-c_1a_2+c_2a_1}=\\frac{1}{a_1b_2+a_2b_1}<\/span><\/span><\/p>\n

\u09ac\u09be, \\frac{x}{\\begin{vmatrix}\n\nc_1 & b_1\\\\\n\nc_2 & b_1\n\n\\end{vmatrix}}=\\frac{y}{\\begin{vmatrix}\n\na_1 & c_1\\\\\n\na_2 & c_2\n\n\\end{vmatrix}}=\\frac{1}{\\begin{vmatrix}\n\na_1 & b_1\\\\\n\na_2 & b_2\n\n\\end{vmatrix}}<\/span><\/span><\/p>\n

\u09ac\u09be, \\frac{x}{D_x}=\\frac{y}{D_y}=\\frac{1}{D}<\/span><\/span><\/p>\n

\\therefore x=\\frac{D_x}{D}, y=\\frac{D_y}{D}<\/span> \u09b9\u09a4\u09c7 <\/span>x,y<\/span>\u00a0 \u098f\u09b0 \u09ae\u09be\u09a8 \u0985\u09b0\u09cd\u09a5\u09be\u09ce, \u09aa\u09cd\u09b0\u09a6\u09a4\u09cd\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u099c\u09cb\u099f\u09c7\u09b0 \u09b8\u09ae\u09be\u09a7\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u09be \u09af\u09be\u09df\u0964\u00a0<\/span><\/p>\n

\u09a4\u09bf\u09a8 \u099a\u09b2\u0995\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u098f\u0995\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u099c\u09cb\u099f\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7-\u00a0\u00a0<\/b><\/h4>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u09aa\u09cd\u09b0\u09a6\u09a4\u09cd\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u099c\u09cb\u099f :<\/span> a_1x+b_1y+c_1z=d_1......(i)\\\\\n\na_2x+b_2y+c_2z=d_2......(ii)\\\\\n\na_3x+b_3y+c_3z=d_3......(iii)<\/span><\/span><\/p>\n

\u2234<\/span> \u0986\u09ae\u09b0\u09be \u09aa\u09be\u0987, \\frac{x}{D_x}=\\frac{y}{D_y}=\\frac{z}{D_z}=\\frac{1}{D}<\/span><\/span><\/p>\n

\u098f\u0996\u09be\u09a8\u09c7, D=\\begin{vmatrix}\n\na_{11} & a_{12} & a_{13}\\\\\n\na_{21} & a_{22} & a_{23}\\\\\n\na_{31} & a_{32} & a_{33}\n\n\\end{vmatrix}\\neq 0;<\/span> <\/span>x,y,z<\/span> \u09b8\u09b9\u0997\u0997\u09c1\u09b2\u09bf \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0997\u09a0\u09bf\u09a4 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995\u0964<\/span><\/p>\n

D_x=\\begin{vmatrix}\n\nd_1 & b_1 & c_1\\\\\n\nd_2 & b_2 & c_2\\\\\n\nd_3 & b_3 & c_3\n\n\\end{vmatrix}<\/span> ; <\/span>x<\/span> \u098f\u09b0 \u09b8\u09b9\u0997\u09c7\u09b0 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09c7 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995 \u09aa\u09a6 \u09a8\u09bf\u09df\u09c7 \u0997\u09a0\u09bf\u09a4 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995\u0964\u00a0<\/span><\/p>\n

\u0986\u09ac\u09be\u09b0 D_y=\\begin{vmatrix}\n\na_1 & d_1 & c_1\\\\\n\na_2 & d_2 & c_2\\\\\n\na_3 & d_3 & c_3\n\n\\end{vmatrix}<\/span><\/span>\u00a0; <\/span>y<\/span> \u098f\u09b0 \u09b8\u09b9\u0997\u09c7\u09b0 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09c7 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995 \u09aa\u09a6 \u09a8\u09bf\u09df\u09c7 \u0997\u09a0\u09bf\u09a4 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995\u0964\u00a0<\/span><\/p>\n

\u098f\u09ac\u0982 D_z=\\begin{vmatrix}\n\na_1 & b_1 & d_1\\\\\n\na_2 & b_2 & d_2\\\\\n\na_3 & b_3 & d_3\n\n\\end{vmatrix}<\/span><\/span>\u00a0;<\/span>z<\/span> \u098f\u09b0 \u09b8\u09b9\u0997\u09c7\u09b0 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09c7 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995 \u09aa\u09a6 \u09a8\u09bf\u09df\u09c7 \u0997\u09a0\u09bf\u09a4 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995\u0964\u00a0<\/span><\/p>\n

\\therefore x=\\frac{D_x}{D}, y=\\frac{D_y}{D}, z=\\frac{D_z}{D}<\/span> \u09b9\u09a4\u09c7 <\/span>x,y,z<\/span>\u00a0 \u098f\u09b0 \u09ae\u09be\u09a8 \u0985\u09b0\u09cd\u09a5\u09be\u09ce, \u09aa\u09cd\u09b0\u09a6\u09a4\u09cd\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u099c\u09cb\u099f\u09c7\u09b0 \u09b8\u09ae\u09be\u09a7\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u09be \u09af\u09be\u09df\u0964\u00a0<\/span><\/p>\n

\u09a6\u09cd\u09b0\u09b7\u09cd\u099f\u09ac\u09cd\u09af : <\/b>\u09af\u09a6\u09bf <\/span>D\u22600<\/span>\u00a0 \u09b9\u09df, \u09a4\u09ac\u09c7 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u099c\u09cb\u099f\u09c7\u09b0 \u0985\u09a8\u09a8\u09cd\u09af \u09b8\u09ae\u09be\u09a7\u09be\u09a8 \u09ac\u09bf\u09a6\u09cd\u09af\u09ae\u09be\u09a8\u0964 \u0995\u09c7\u09ac\u09b2 <\/span>D\u22600<\/span>\u00a0 \u09b6\u09b0\u09cd\u09a4\u09c7\u0987 \u0995\u09cd\u09b0\u09c7\u09ae\u09be\u09b0\u09c7\u09b0 \u09a8\u09bf\u09df\u09ae \u09aa\u09cd\u09b0\u09af\u09cb\u099c\u09cd\u09af\u0964\u00a0\u00a0\u00a0<\/span><\/p>\n

\u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8\u09c7\u09b0 \u09b8\u09be\u09b9\u09be\u09af\u09cd\u09af\u09c7 \u098f\u0995\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u00a0 \u099c\u09cb\u099f\u09c7\u09b0 \u09b8\u09ae\u09be\u09a7\u09be\u09a8 (Solution of system of linear equations using inverse matrix)<\/b><\/span><\/h3>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u09aa\u09cd\u09b0\u09a6\u09a4\u09cd\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u099c\u09cb\u099f: a_1x+b_1y+c_1z=d_1......(i)\\\\\n\na_2x+b_2y+c_2z=d_2......(ii)\\\\\n\na_3x+b_3y+c_3z=d_3......(iii)<\/span><\/span><\/p>\n

\u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u099c\u09cb\u099f\u099f\u09bf \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 \u0986\u0995\u09be\u09b0\u09c7 \u09b2\u09bf\u0996\u09c7 \u09aa\u09be\u0987, \\begin{bmatrix}\n\na_1 & b_1 & c_1\\\\\n\na_2 & b_2 & c_2\\\\\n\na_3 & b_3 & c_3\n\n\\end{bmatrix}\\begin{bmatrix}\n\nx\\\\\n\ny\\\\\n\nz\n\n\\end{bmatrix}=\\begin{bmatrix}\n\nd_1\\\\\n\nd_2\\\\\n\nd_3\n\n\\end{bmatrix}<\/span><\/span><\/p>\n

\\Rightarrow AX=B<\/span> \u09af\u09c7\u0996\u09be\u09a8\u09c7\u00a0 A=\\begin{bmatrix}\n\na_1 & b_1 & c_1\\\\\n\na_2 & b_2 & c_2\\\\\n\na_3 & b_3 & c_3\n\n\\end{bmatrix}<\/span><\/span>\u00a0\u098f\u0995\u099f\u09bf \u0985\u09ac\u09cd\u09af\u09a4\u09bf\u0995\u09cd\u09b0\u09ae\u09c0 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8, X=\\begin{bmatrix}\n\nx\\\\\n\ny\\\\\n\nz\n\n\\end{bmatrix}<\/span><\/span>\u00a0\u098f\u09ac\u0982 B=\\begin{bmatrix}\n\nd_1\\\\\n\nd_2\\\\\n\nd_3\n\n\\end{bmatrix}<\/span><\/span><\/p>\n

\\therefore X=A^{-1}B<\/span> \u09b9\u09a4\u09c7 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 \u09b8\u09ae\u09a4\u09be \u09aa\u09cd\u09b0\u09df\u09cb\u0997 \u0995\u09b0\u09c7 <\/span>x,y,z<\/span> \u09ae\u09be\u09a8 \u09aa\u09be\u0993\u09df\u09be \u09af\u09be\u09df\u0964<\/span><\/p>\n

 <\/p>\n

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 <\/p>\n

\u098f\u0987\u099a\u098f\u09b8\u09b8\u09bf \u0993 \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u09aa\u09b0\u09c0\u0995\u09cd\u09b7\u09be\u09b0\u09cd\u09a5\u09c0\u09a6\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u0986\u09ae\u09be\u09a6\u09c7\u09b0 \u0995\u09cb\u09b0\u09cd\u09b8\u09b8\u09ae\u09c2\u09b9\u0983<\/strong><\/em><\/p>\n