Problem to solve
Solve \[x-y+3z=2\], \[2x-3y=-4\], \[x+y+z=-2\]
Solution
STEP 1
Solve the following system: {x - y + 3 z = 2 -3 y + 2 x = -4 x + y + z = -2
STEP 2
Express the system in matrix form: (1 | -1 | 3 2 | -3 | 0 1 | 1 | 1)(x y z) = (2 -4 -2)
STEP 3
Solve the system with Cramer's rule: x = 2 | -1 | 3 -4 | -3 | 0 -2 | 1 | 1/1 | -1 | 3 2 | -3 | 0 1 | 1 | 1 and y = 1 | 2 | 3 2 | -4 | 0 1 | -2 | 1/1 | -1 | 3 2 | -3 | 0 1 | 1 | 1 and z = 1 | -1 | 2 2 | -3 | -4 1 | 1 | -2/1 | -1 | 3 2 | -3 | 0 1 | 1 | 1
STEP 4
Evaluate the determinant 1 | -1 | 3 2 | -3 | 0 1 | 1 | 1 = 14: x = 2 | -1 | 3 -4 | -3 | 0 -2 | 1 | 1/14 and y = 1 | 2 | 3 2 | -4 | 0 1 | -2 | 1/14 and z = 1 | -1 | 2 2 | -3 | -4 1 | 1 | -2/14
STEP 5
Evaluate the determinant 2 | -1 | 3 -4 | -3 | 0 -2 | 1 | 1 = -40: x = (-40)/14 and y = 1 | 2 | 3 2 | -4 | 0 1 | -2 | 1/14 and z = 1 | -1 | 2 2 | -3 | -4 1 | 1 | -2/14
STEP 6
The gcd of -40 and 14 is 2, so (-40)/14 = (-20×2)/(7×2) = -20/7: x = -20/7 and y = 1 | 2 | 3 2 | -4 | 0 1 | -2 | 1/14 and z = 1 | -1 | 2 2 | -3 | -4 1 | 1 | -2/14
STEP 7
Evaluate the determinant 1 | 2 | 3 2 | -4 | 0 1 | -2 | 1 = -8: x = (-20)/7 and y = (-8)/14 and z = 1 | -1 | 2 2 | -3 | -4 1 | 1 | -2/14
STEP 8
The gcd of -8 and 14 is 2, so (-8)/14 = (-4×2)/(7×2) = -4/7: x = (-20)/7 and y = -4/7 and z = 1 | -1 | 2 2 | -3 | -4 1 | 1 | -2/14
STEP 9
Evaluate the determinant 1 | -1 | 2 2 | -3 | -4 1 | 1 | -2 = 20: x = (-20)/7 and y = (-4)/7 and z = 20/14
STEP 10
The gcd of 20 and 14 is 2, so 20/14 = (10×2)/(7×2) = 10/7: Answer: | | x = (-20)/7 and y = (-4)/7 and z = 10/7
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