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Existence of Solution in Linear Equations Example LN1 2x+3y=4, 3x+4y=5, x+5y=9, 6x+7y=8 ⎡2 ⎢3 ⎢ ⎢1 ⎢ ⎣6 3 4 5 7 4⎤ 5⎥⎥ 9⎥ ⎥ 8⎦ ⎡2 ⎢0 ⎢ ⎢0 ⎢ ⎣0 -1.5R1+R2→R2 -0.5R1+R3→R3 -3R1+R4→R4 4 ⎤ ⎡2 3 ⎢ 7R2+R3→R3 0 - 0.5 - 1⎥⎥ ⎢ -4R2+R4→R4 ⎢0 0 0⎥ ⎢ ⎥ 0⎦ ⎣0 0 ⎡2 3 ⎤ ⎡2 3 ⎢0 - 0.5 ⎥ ⎢0 - 0.5 __ ⎥, A= ⎢ A= ⎢ ⎢0 0 ⎥ ⎢0 0 ⎢ ⎥ ⎢ ⎣0 0 ⎦ ⎣0 0 rank A=2 __ rank A =2 3 4⎤ - 0.5 -1⎥⎥ 3.5 7⎥ ⎥ - 2 - 4⎦ 4 ⎤ - 1⎥⎥ 0⎥ ⎥ 0⎦ n=2 __ __ rank A =3 n=2 __ rankA≠ rank A → No solution ------------------------------------------------------. Example LN3 2x+3y=4, 6x+9y=12, 8+12y=16, 4x+6y=8 ⎡2 3 4⎤ ⎡2 3 4 ⎤ ⎢6 9 12 ⎥ ⎢0 0 0 ⎥ Row ⎢ ⎥ ⎥ Operations ⎢ ⎢ ⎢8 12 16⎥ 0 0 0⎥ → ⎢ ⎥ ⎢ ⎥ ⎣4 6 8 ⎦ ⎣0 0 0 ⎦ rank A=1 __ rank A =1 __ 2x+3y+4z-w=-5 6x+9y+12z-3w=21 ⎡2 3 4 -1 - 5⎤ ⎡2 3 4 -1 - 5 ⎤ ⎢6 9 12 - 3 21⎥ → ⎢0 0 0 0 36⎥ ⎣ ⎦ ⎣ ⎦ n=4 __ rank A =2 rankA=1 __ rankA ≠ rank A Example LN5 → No solution 2x+3y+4z-w=-5 4x+2y+5z-3w=-3 rank A= rank A =n → Unique Solution Notice: Number of equations has no importance ------------------------------------------------------. Example LN2 2x+3y=4, 3x+4y=5, x+5y=9, 6x+7y=2 4 ⎤ ⎡2 3 4⎤ ⎡2 3 Row ⎢3 4 5⎥ ⎢0 - 0.5 - 1 ⎥ ⎢ ⎥ Operations ⎢ ⎥ → ⎢1 5 9 ⎥ ⎢0 0 - 6⎥ ⎥ ⎢ ⎢ ⎥ 0 ⎦ ⎣6 7 2⎦ ⎣0 0 4 ⎤ ⎡2 3 ⎡2 3 ⎤ ⎢0 - 0.5 ⎥ ⎢ __ 0 - 0.5 - 1 ⎥⎥ ⎢ ⎥ ⎢ A= , A= ⎢0 0 ⎢0 0 ⎥ - 6⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎦ ⎣0 0 ⎦ ⎣0 0 rank A=2 Example LN4 n=2 rankA= rank A =r r<n → Multiple Solution. n-r=2-1=1. 1 variable free Solution: Take the echolen form equation 2x+3 y=4. Set x freely and calculate y Set x=1 → 2 1+3 y=4 → y=2/3=0.666 Set x=0 → y=4/3=1.3333 Set x=-5.5 → y=5 ⎡2 3 4 -1 - 5 ⎤ ⎡2 3 4 -1 - 5⎤ ⎢4 2 5 - 3 - 3⎥ → ⎢0 - 4 - 3 -1 7⎥ ⎣ ⎦ ⎣ ⎦ __ n=4 rankA=rank A =r=2 r<n multiple solution. n-r =4-2 =2 2 variable free To solve x and y, we use echolen form equations. 2x+3y+4z-w=-5 -4y-3z-w=7 ---------------------------------Set z=0 w=1 (arbitrary selection) 2x+3y+4 0 -1=-5 -4y - 3 0 - 1=7 → y=-2 replace y=-2 z=0 w=1 into the first equation 2x+3 (-2) -1=-5 → x=1 Solution 1=[ x=1 y=-2 z=0 w=1 ] ------------------- ------------------Set Set z=2, w=3 (arbitrary selection) 2x+3y +4 2 - 3=-5 → 2x+3y=-10 -4y-3 2 - 3 =7 → -4y =16 → y= -4 from first equation 2x+3y=-10 → 2x+3(-4)=-10 → x=1 Solution 2=[ x=1 y=-4 z=2 w=3 ] ---------------------------------Set Set z=0, w=0 (arbitrary selection) 2x+3y +4 0 - 0=-5 → 2x+3y=-5 -4y-3 0 - 0 =7 → -4y =7 → y= -1.75 from first equation 2x+3y=-5 → 2x+3(-1.75)=-5 → x=0.125 Solution 3=[ x=0.125 y=-1.75 z=0 w=0 ] Example LN6 2x+3y+4z-w=-5, ⎡2 3 ⎢4 2 ⎢ ⎢6 1 ⎢ ⎢- 2 5 ⎢10 3 ⎣ 6x+y+6z-5w =-1 10x+3y+11z -8w=-4 4x+2y+5z-3w=-3, -2x+5y+2z+3w=-9, 4 -1 - 5 ⎤ ⎡2 3 4 - 1 5 - 3 - 3 ⎥⎥ ⎢0 - 4 - 3 - 1 6 - 5 -1 ⎥ → ⎢ ⎢0 0 0 0 ⎥ 2 3 -9⎥ ⎢ 0 ⎢0 0 0 11 - 8 - 4⎥⎦ ⎢⎣ 0 0 0 0 - 5⎤ 7 ⎥⎥ 0⎥ ⎥ 0⎥ 0 ⎥⎦ __ n=4 rankA=rank A =r=2 r<n multiple solution. n-r =4-2 =2 2 variable free Echolen form equations are 2x+3y+4z-w=-5 -4y-3z-w=7 These equations are solved in Example LN5