Problem to solve
Solve \[\frac{x}{2}-3z=-2\], \[3y-\frac{z}{2}=2\], \[x+3y-z=5\]
Solution
STEP 1
Solve the following system: {-3 z + x/2 = -2 -z/2 + 3 y = 2 x + 3 y - z = 5
STEP 2
Hint: | Choose an equation and a variable to solve for. In the third equation, look to solve for x: {-3 z + x/2 = -2 -z/2 + 3 y = 2 x + 3 y - z = 5
STEP 3
Hint: | Solve for x. Subtract -z + 3 y from both sides: {-3 z + x/2 = -2 -z/2 + 3 y = 2 x = z - 3 y + 5
STEP 4
Hint: | Perform a substitution. Substitute x = z - 3 y + 5 into the first equation: {1/2 (z - 3 y + 5) - 3 z = -2 -z/2 + 3 y = 2 x = z - 3 y + 5
STEP 5
Hint: | Expand the left hand side of the equation 1/2 (z - 3 y + 5) - 3 z = -2. (z - 3 y + 5)/2 - 3 z = (z/2 - (3 y)/2 + 5/2) - 3 z = -(5 z)/2 - (3 y)/2 + 5/2: {(-(5 z)/2 - (3 y)/2 + 5/2) = -2 -z/2 + 3 y = 2 x = z - 3 y + 5
STEP 6
Hint: | Choose an equation and a variable to solve for. In the first equation, look to solve for y: {-(5 z)/2 - (3 y)/2 + 5/2 = -2 -z/2 + 3 y = 2 x = z - 3 y + 5
STEP 7
Hint: | Isolate terms with y to the left hand side. Subtract -(5 z)/2 + 5/2 from both sides: {-(3 y)/2 = (5 z)/2 - 9/2 -z/2 + 3 y = 2 x = z - 3 y + 5
STEP 8
Hint: | Solve for y. Multiply both sides by -2/3: {y = -(5 z)/3 + 3 -z/2 + 3 y = 2 x = z - 3 y + 5
STEP 9
Hint: | Perform a substitution. Substitute y = -(5 z)/3 + 3 into the second equation: {y = -(5 z)/3 + 3 3 (-(5 z)/3 + 3) - z/2 = 2 x = z - 3 y + 5
STEP 10
Hint: | Expand the left hand side of the equation 3 (-(5 z)/3 + 3) - z/2 = 2. 3 (-(5 z)/3 + 3) - z/2 = -z/2 + (-5 z + 9) = -(11 z)/2 + 9: {y = -(5 z)/3 + 3 (-(11 z)/2 + 9) = 2 x = z - 3 y + 5
STEP 11
Hint: | Choose an equation and a variable to solve for. In the second equation, look to solve for z: {y = -(5 z)/3 + 3 -(11 z)/2 + 9 = 2 x = z - 3 y + 5
STEP 12
Hint: | Isolate terms with z to the left hand side. Subtract 9 from both sides: {y = -(5 z)/3 + 3 -(11 z)/2 = -7 x = z - 3 y + 5
STEP 13
Hint: | Solve for z. Multiply both sides by -2/11: {y = -(5 z)/3 + 3 z = 14/11 x = z - 3 y + 5
STEP 14
Hint: | Perform a back substitution. Substitute z = 14/11 into the first and third equations: {y = 29/33 z = 14/11 x = -3 y + 69/11
STEP 15
Hint: | Perform a back substitution. Substitute y = 29/33 into the third equation: {y = 29/33 z = 14/11 x = 40/11
STEP 16
Hint: | Sort results. Collect results in alphabetical order: Answer: | | {x = 40/11 y = 29/33 z = 14/11
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