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Solve the system of equations by elimination.\newline2x−y−z=−42x - y - z = -4\newline−2x−3y−2z=−16-2x - 3y - 2z = -16\newline2x+y−2z=162x + y - 2z = 16

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Solve a system of equations in three variables using eliminationright chevron icon

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Q. Solve the system of equations by elimination.\newline2x−y−z=−42x - y - z = -4\newline−2x−3y−2z=−16-2x - 3y - 2z = -16\newline2x+y−2z=162x + y - 2z = 16
  1. Eliminate x by adding equations: Add the first and second equations to eliminate x.\newline(2x−y−z)+(−2x−3y−2z)=−4+(−16)(2x - y - z) + (-2x - 3y - 2z) = -4 + (-16)\newline2x−y−z−2x−3y−2z=−202x - y - z - 2x - 3y - 2z = -20\newline−4y−3z=−20-4y - 3z = -20
  2. Eliminate xx by adding equations: Add the first and third equations to eliminate xx.
    (2x−y−z)+(2x+y−2z)=−4+16(2x - y - z) + (2x + y - 2z) = -4 + 16
    2x−y−z+2x+y−2z=122x - y - z + 2x + y - 2z = 12
    4x−3z=124x - 3z = 12
  3. Solve for x: Divide the last equation by 44 to solve for xx. \newline4x−3z=124x - 3z = 12\newlinex−(34)z=3x - \left(\frac{3}{4}\right)z = 3\newlinex=3+(34)zx = 3 + \left(\frac{3}{4}\right)z
  4. Substitute xx into first equation: Substitute x=3+34zx = 3 + \frac{3}{4}z into the first equation.\newline2(3+34z)−y−z=−42(3 + \frac{3}{4}z) - y - z = -4\newline6+32z−y−z=−46 + \frac{3}{2}z - y - z = -4\newline6+12z−y=−46 + \frac{1}{2}z - y = -4
  5. Isolate terms with variables: Subtract 66 from both sides to isolate the terms with variables.\newline6+(12)z−y−6=−4−66 + \left(\frac{1}{2}\right)z - y - 6 = -4 - 6\newline(12)z−y=−10\left(\frac{1}{2}\right)z - y = -10
  6. Get rid of fraction: Multiply the equation by 22 to get rid of the fraction.2(12z−y)=2(−10)2\left(\frac{1}{2}z - y\right) = 2(-10)z−2y=−20z - 2y = -20
  7. Align y terms: Now we have a system with two equations and two variables:\newline−4y−3z=−20-4y - 3z = -20\newlinez−2y=−20z - 2y = -20\newlineMultiply the second equation by 22 to align the y terms.\newline2(z−2y)=2(−20)2(z - 2y) = 2(-20)\newline2z−4y=−402z - 4y = -40
  8. Eliminate y: Add the modified second equation to the first equation to eliminate y.\newline(−4y−3z)+(2z−4y)=−20+(−40)(-4y - 3z) + (2z - 4y) = -20 + (-40)\newline−8y−z=−60-8y - z = -60
  9. Solve for y: Divide the equation by −8-8 to solve for y.\newline−8y−z=−60-8y - z = -60\newliney+18z=7.5y + \frac{1}{8}z = 7.5\newliney=7.5−18zy = 7.5 - \frac{1}{8}z
  10. Substitute yy into equation: Substitute y=7.5−18zy = 7.5 - \frac{1}{8}z into the equation z−2y=−20z - 2y = -20.\newlinez−2(7.5−18z)=−20z - 2(7.5 - \frac{1}{8}z) = -20\newlinez−15+14z=−20z - 15 + \frac{1}{4}z = -20\newline54z=−5\frac{5}{4}z = -5\newlinez=−554z = \frac{-5}{\frac{5}{4}}\newlinez=−4z = -4
  11. Find zz: Substitute z=−4z = -4 into y=7.5−18zy = 7.5 - \frac{1}{8}z to find yy.\newliney=7.5−18(−4)y = 7.5 - \frac{1}{8}(-4)\newliney=7.5+0.5y = 7.5 + 0.5\newliney=8y = 8
  12. Find yy: Substitute z=−4z = -4 into x=3+(34)zx = 3 + \left(\frac{3}{4}\right)z to find xx.\newlinex=3+(34)(−4)x = 3 + \left(\frac{3}{4}\right)(-4)\newlinex=3−3x = 3 - 3\newlinex=0x = 0

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