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
Hint: | Choose an equation and a variable to solve for. In the second equation, look to solve for y: {2 x - y/3 + 2 z = -1 x + y + z/2 = -1 x - 2 y + z = 2
STEP 3
Hint: | Solve for y. Subtract z/2 + x from both sides: {2 x - y/3 + 2 z = -1 y = (-z/2 - x) - 1 x - 2 y + z = 2
STEP 4
Hint: | Perform a substitution. Substitute y = -z/2 - x - 1 into the first and third equations: {2 x + 1/3 (z/2 + x + 1) + 2 z = -1 y = -z/2 - x - 1 x - 2 (-z/2 - x - 1) + z = 2
STEP 5
Hint: | Expand the left hand side of the equation 2 x + 1/3 (z/2 + x + 1) + 2 z = -1. 2 x + (z/2 + x + 1)/3 + 2 z = 2 x + (z/6 + x/3 + 1/3) + 2 z = (13 z)/6 + (7 x)/3 + 1/3: {((13 z)/6 + (7 x)/3 + 1/3) = -1 y = -z/2 - x - 1 x - 2 (-z/2 - x - 1) + z = 2
STEP 6
Hint: | Expand the left hand side of the equation x - 2 (-z/2 - x - 1) + z = 2. x - 2 (-z/2 - x - 1) + z = x + (z + 2 x + 2) + z = 2 z + 3 x + 2: {(13 z)/6 + (7 x)/3 + 1/3 = -1 y = -z/2 - x - 1 (2 z + 3 x + 2) = 2
STEP 7
Hint: | Choose an equation and a variable to solve for. In the first equation, look to solve for x: {(13 z)/6 + (7 x)/3 + 1/3 = -1 y = -z/2 - x - 1 2 z + 3 x + 2 = 2
STEP 8
Hint: | Isolate terms with x to the left hand side. Subtract (13 z)/6 + 1/3 from both sides: {(7 x)/3 = -(13 z)/6 - 4/3 y = -z/2 - x - 1 2 z + 3 x + 2 = 2
STEP 9
Hint: | Solve for x. Multiply both sides by 3/7: {x = -(13 z)/14 - 4/7 y = -z/2 - x - 1 2 z + 3 x + 2 = 2
STEP 10
Hint: | Perform a substitution. Substitute x = -(13 z)/14 - 4/7 into the third equation: {x = -(13 z)/14 - 4/7 y = -z/2 - x - 1 2 + 3 (-(13 z)/14 - 4/7) + 2 z = 2
STEP 11
Hint: | Expand the left hand side of the equation 2 + 3 (-(13 z)/14 - 4/7) + 2 z = 2. 2 + 3 (-(13 z)/14 - 4/7) + 2 z = 2 z + (-(39 z)/14 - 12/7) + 2 = -(11 z)/14 + 2/7: {x = -(13 z)/14 - 4/7 y = -z/2 - x - 1 (-(11 z)/14 + 2/7) = 2
STEP 12
Hint: | Choose an equation and a variable to solve for. In the third equation, look to solve for z: {x = -(13 z)/14 - 4/7 y = -z/2 - x - 1 -(11 z)/14 + 2/7 = 2
STEP 13
Hint: | Isolate terms with z to the left hand side. Subtract 2/7 from both sides: {x = -(13 z)/14 - 4/7 y = -z/2 - x - 1 -(11 z)/14 = 12/7
STEP 14
Hint: | Solve for z. Multiply both sides by -14/11: {x = -(13 z)/14 - 4/7 y = -z/2 - x - 1 z = -24/11
STEP 15
Hint: | Perform a back substitution. Substitute z = -24/11 into the first and second equations: {x = 16/11 y = -x + 1/11 z = -24/11
STEP 16
Hint: | Perform a back substitution. Substitute x = 16/11 into the second equation: Answer: | | {x = 16/11 y = -15/11 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