Problem to solve
Solve \[2x-\frac{y}{3}+2z=-1\], \[x+y+\frac{z}{2}=-1\], \[x-2y+z=2\]
Solution
STEP 1
Solve the following system: {2 x - y/3 + 2 z = -1 x + y + z/2 = -1 x - 2 y + z = 2
STEP 2
Express the system in matrix form: (2 | -1/3 | 2 1 | 1 | 1/2 1 | -2 | 1)(x y z) = (-1 -1 2)
STEP 3
Solve the system with Cramer's rule: x = (-1 | -1/3 | 2 -1 | 1 | 1/2 2 | -2 | 1)/(2 | -1/3 | 2 1 | 1 | 1/2 1 | -2 | 1) and y = (2 | -1 | 2 1 | -1 | 1/2 1 | 2 | 1)/(2 | -1/3 | 2 1 | 1 | 1/2 1 | -2 | 1) and z = (2 | -1/3 | -1 1 | 1 | -1 1 | -2 | 2)/(2 | -1/3 | 2 1 | 1 | 1/2 1 | -2 | 1)
STEP 4
Evaluate the determinant 2 | -1/3 | 2 1 | 1 | 1/2 1 | -2 | 1 = (-11)/6: x = (-1 | -1/3 | 2 -1 | 1 | 1/2 2 | -2 | 1)/(-11/6) and y = (2 | -1 | 2 1 | -1 | 1/2 1 | 2 | 1)/(-11/6) and z = (2 | -1/3 | -1 1 | 1 | -1 1 | -2 | 2)/(-11/6)
STEP 5
The gcd of -1 | -1/3 | 2 -1 | 1 | 1/2 2 | -2 | 1 and -11/6 is 1/6, so (-1 | -1/3 | 2 -1 | 1 | 1/2 2 | -2 | 1)/((-11)/6) = (6 -1 | -1/3 | 2 -1 | 1 | 1/2 2 | -2 | 1)/(6 ((-11)/6)) = -6/11 -1 | -1/3 | 2 -1 | 1 | 1/2 2 | -2 | 1: x = -6/11 -1 | -1/3 | 2 -1 | 1 | 1/2 2 | -2 | 1 and y = (2 | -1 | 2 1 | -1 | 1/2 1 | 2 | 1)/((-11)/6) and z = (2 | -1/3 | -1 1 | 1 | -1 1 | -2 | 2)/((-11)/6)
STEP 6
The gcd of 2 | -1 | 2 1 | -1 | 1/2 1 | 2 | 1 and -11/6 is 1/6, so (2 | -1 | 2 1 | -1 | 1/2 1 | 2 | 1)/((-11)/6) = (6 2 | -1 | 2 1 | -1 | 1/2 1 | 2 | 1)/(6 ((-11)/6)) = -6/11 2 | -1 | 2 1 | -1 | 1/2 1 | 2 | 1: x = (-6 -1 | -1/3 | 2 -1 | 1 | 1/2 2 | -2 | 1)/11 and y = -6/11 2 | -1 | 2 1 | -1 | 1/2 1 | 2 | 1 and z = (2 | -1/3 | -1 1 | 1 | -1 1 | -2 | 2)/((-11)/6)
STEP 7
The gcd of 2 | -1/3 | -1 1 | 1 | -1 1 | -2 | 2 and -11/6 is 1/6, so (2 | -1/3 | -1 1 | 1 | -1 1 | -2 | 2)/((-11)/6) = (6 2 | -1/3 | -1 1 | 1 | -1 1 | -2 | 2)/(6 ((-11)/6)) = -6/11 2 | -1/3 | -1 1 | 1 | -1 1 | -2 | 2: x = (-6 -1 | -1/3 | 2 -1 | 1 | 1/2 2 | -2 | 1)/11 and y = (-6 2 | -1 | 2 1 | -1 | 1/2 1 | 2 | 1)/11 and z = -6/11 2 | -1/3 | -1 1 | 1 | -1 1 | -2 | 2
STEP 8
Evaluate the determinant -1 | -1/3 | 2 -1 | 1 | 1/2 2 | -2 | 1 = (-8)/3: x = (-6)/11×-8/3 and y = (-6 2 | -1 | 2 1 | -1 | 1/2 1 | 2 | 1)/11 and z = (-6 2 | -1/3 | -1 1 | 1 | -1 1 | -2 | 2)/11
STEP 9
(-6)/11×(-8)/3 = 16/11: x = 16/11 and y = (-6 2 | -1 | 2 1 | -1 | 1/2 1 | 2 | 1)/11 and z = (-6 2 | -1/3 | -1 1 | 1 | -1 1 | -2 | 2)/11
STEP 10
Evaluate the determinant 2 | -1 | 2 1 | -1 | 1/2 1 | 2 | 1 = 5/2: x = 16/11 and y = (-6)/11×5/2 and z = (-6 2 | -1/3 | -1 1 | 1 | -1 1 | -2 | 2)/11
STEP 11
(-6)/11×5/2 = (-15)/11: x = 16/11 and y = -15/11 and z = (-6 2 | -1/3 | -1 1 | 1 | -1 1 | -2 | 2)/11
STEP 12
Evaluate the determinant 2 | -1/3 | -1 1 | 1 | -1 1 | -2 | 2 = 4: x = 16/11 and y = (-15)/11 and z = (-6)/11×4
STEP 13
(-6)/11×4 = (-24)/11: Answer: | | x = 16/11 and y = (-15)/11 and z = -24/11
More problems
Number
001Solve \(3x+2y=5\), \(-4x+5y=2\)
002Solve \(-4x+5y+3=x+4\), \(6x-7y=2y-1\)
003Solve \(-5x+2y=-3+2x\), \(4x-2y+3=7-4x\)
004Solve \(3y+5x+1-2y=2y-3\), \(3y-2+5x=2x-3y+2\)
005Solve \(3x+5y=2\), \(-5x+6y=5\)
006Solve \(-\frac{2}{3}x+5y=4\), \(7x-5y=4\)
007Solve \( 4x-5y=3\), \(-2x+6y=-2\)
008Solve \(\frac{6}{5}x+\frac{3}{2}y=5\), \(-\frac{3}{5}x+y=5y+2\)
009Solve \(\frac{3}{2}x-\frac{1}{2}y=6x-2\), \(3x+\frac{2}{3}y=-2y-1\)
010Solve \(-2x+6y-4x=5x-3y+3\), \(7y+4x=9-2y\)
011Solve \(\frac{2x+y}{3}-\frac{2(x+y)}{4}=2\), \(-\frac{2(x-y)}{5}-\frac{x+2y}{2}=-1\)
012Solve \(-\frac{2(x-y)}{5}+\frac{2x-y}{2}=-1\), \(4(2x+3)-\frac{3x-4y}{2}=5\)
013Solve \(\frac{x+2}{2}+\frac{3y-1}{4}=-1\), \(2x+y-2(3x-4y)=x+3\)
014Solve \(\frac{2x-1}{4}+\frac{x-2y}{3}=4+x\), \(\frac{x+y}{3}-(x-y)=-2y+1\)
015Solve \(3(x-5y)+2(2x-1)=2(x-y)\), \(2(x-1)-3(x-y)=-3x\)
016Solve \(x-y+3z=2\), \(2x-3y=-4\), \(x+y+z=-2\)
017Solve \(-4x+y-5z=2\), \(4x+2y+z=1\), \(2x-y+3z=4\)
018Solve \(-10x+3y+z=-2\), \(3x-2y+5z=4\), \(-6x+4y+2z=1\)
019Solve \(4x-3y=7\), \(2x-3y+z=7\), \(4x-y+z=-8\)
020Solve \(-x+3y+2z=5\), \(-3x-2y+z=-4\), \(x-y+z=3\)
021Solve \(\frac{x}{3}-\frac{y}{2}+z=2\), \(-x-\frac{y}{3}+2z=3\), \(2x-3y+\frac{z}{3}=-1\)
022Solve \(2x-\frac{y}{3}+2z=-1\), \(x+y+\frac{z}{2}=-1\), \(x-2y+z=2\)
023Solve \(-\frac{3x}{2}+3y+z=2\), \(3x-y+z=5\), \(\frac{x}{3}-y+z=2\)
024Solve \(\frac{2x}{3}-2z=-2\), \(2y-4z=2\), \(\frac{x}{3}-5y=2\)
025Solve \(\frac{x}{2}-3z=-2\), \(3y-\frac{z}{2}=2\), \(x+3y-z=5\)
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Edited by tododemate® - 2021