Problem to solve
Solve \[\frac{x}{3}-\frac{y}{2}+z=2\], \[-x-\frac{y}{3}+2z=3\], \[2x-3y+\frac{z}{3}=-1\]
Solution
STEP 1
Solve the following system: {x/3 - y/2 + z = 2 -x - y/3 + 2 z = 3 2 x - 3 y + z/3 = -1
STEP 2
Express the system in matrix form: (1/3 | -1/2 | 1 -1 | -1/3 | 2 2 | -3 | 1/3)(x y z) = (2 3 -1)
STEP 3
Solve the system with Cramer's rule: x = (2 | -1/2 | 1 3 | -1/3 | 2 -1 | -3 | 1/3)/(1/3 | -1/2 | 1 -1 | -1/3 | 2 2 | -3 | 1/3) and y = (1/3 | 2 | 1 -1 | 3 | 2 2 | -1 | 1/3)/(1/3 | -1/2 | 1 -1 | -1/3 | 2 2 | -3 | 1/3) and z = (1/3 | -1/2 | 2 -1 | -1/3 | 3 2 | -3 | -1)/(1/3 | -1/2 | 1 -1 | -1/3 | 2 2 | -3 | 1/3)
STEP 4
Evaluate the determinant 1/3 | -1/2 | 1 -1 | -1/3 | 2 2 | -3 | 1/3 = 187/54: x = (2 | -1/2 | 1 3 | -1/3 | 2 -1 | -3 | 1/3)/(187/54) and y = (1/3 | 2 | 1 -1 | 3 | 2 2 | -1 | 1/3)/(187/54) and z = (1/3 | -1/2 | 2 -1 | -1/3 | 3 2 | -3 | -1)/(187/54)
STEP 5
The gcd of 2 | -1/2 | 1 3 | -1/3 | 2 -1 | -3 | 1/3 and 187/54 is 1/54, so (2 | -1/2 | 1 3 | -1/3 | 2 -1 | -3 | 1/3)/(187/54) = (54 2 | -1/2 | 1 3 | -1/3 | 2 -1 | -3 | 1/3)/(54×187/54) = 54/187 2 | -1/2 | 1 3 | -1/3 | 2 -1 | -3 | 1/3: x = 54/187 2 | -1/2 | 1 3 | -1/3 | 2 -1 | -3 | 1/3 and y = (1/3 | 2 | 1 -1 | 3 | 2 2 | -1 | 1/3)/(187/54) and z = (1/3 | -1/2 | 2 -1 | -1/3 | 3 2 | -3 | -1)/(187/54)
STEP 6
The gcd of 1/3 | 2 | 1 -1 | 3 | 2 2 | -1 | 1/3 and 187/54 is 1/54, so (1/3 | 2 | 1 -1 | 3 | 2 2 | -1 | 1/3)/(187/54) = (54 1/3 | 2 | 1 -1 | 3 | 2 2 | -1 | 1/3)/(54×187/54) = 54/187 1/3 | 2 | 1 -1 | 3 | 2 2 | -1 | 1/3: x = (54 2 | -1/2 | 1 3 | -1/3 | 2 -1 | -3 | 1/3)/187 and y = 54/187 1/3 | 2 | 1 -1 | 3 | 2 2 | -1 | 1/3 and z = (1/3 | -1/2 | 2 -1 | -1/3 | 3 2 | -3 | -1)/(187/54)
STEP 7
The gcd of 1/3 | -1/2 | 2 -1 | -1/3 | 3 2 | -3 | -1 and 187/54 is 1/54, so (1/3 | -1/2 | 2 -1 | -1/3 | 3 2 | -3 | -1)/(187/54) = (54 1/3 | -1/2 | 2 -1 | -1/3 | 3 2 | -3 | -1)/(54×187/54) = 54/187 1/3 | -1/2 | 2 -1 | -1/3 | 3 2 | -3 | -1: x = (54 2 | -1/2 | 1 3 | -1/3 | 2 -1 | -3 | 1/3)/187 and y = (54 1/3 | 2 | 1 -1 | 3 | 2 2 | -1 | 1/3)/187 and z = 54/187 1/3 | -1/2 | 2 -1 | -1/3 | 3 2 | -3 | -1
STEP 8
Evaluate the determinant 2 | -1/2 | 1 3 | -1/3 | 2 -1 | -3 | 1/3 = 71/18: x = 54/187×71/18 and y = (54 1/3 | 2 | 1 -1 | 3 | 2 2 | -1 | 1/3)/187 and z = (54 1/3 | -1/2 | 2 -1 | -1/3 | 3 2 | -3 | -1)/187
STEP 9
54/187×71/18 = 213/187: x = 213/187 and y = (54 1/3 | 2 | 1 -1 | 3 | 2 2 | -1 | 1/3)/187 and z = (54 1/3 | -1/2 | 2 -1 | -1/3 | 3 2 | -3 | -1)/187
STEP 10
Evaluate the determinant 1/3 | 2 | 1 -1 | 3 | 2 2 | -1 | 1/3 = 14/3: x = 213/187 and y = 54/187×14/3 and z = (54 1/3 | -1/2 | 2 -1 | -1/3 | 3 2 | -3 | -1)/187
STEP 11
54/187×14/3 = 252/187: x = 213/187 and y = 252/187 and z = (54 1/3 | -1/2 | 2 -1 | -1/3 | 3 2 | -3 | -1)/187
STEP 12
Evaluate the determinant 1/3 | -1/2 | 2 -1 | -1/3 | 3 2 | -3 | -1 = 143/18: x = 213/187 and y = 252/187 and z = 54/187×143/18
STEP 13
54/187×143/18 = 39/17: Answer: | | x = 213/187 and y = 252/187 and z = 39/17
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Number
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006Solve \(-\frac{2}{3}x+5y=4\), \(7x-5y=4\)
007Solve \( 4x-5y=3\), \(-2x+6y=-2\)
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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