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Solve the system of equations by elimination.\newlinex−2y−2z=7x - 2y - 2z = 7\newline3x−2y−z=−53x - 2y - z = -5\newline2x−2y+2z=82x - 2y + 2z = 8

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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.\newlinex−2y−2z=7x - 2y - 2z = 7\newline3x−2y−z=−53x - 2y - z = -5\newline2x−2y+2z=82x - 2y + 2z = 8
  1. Eliminate y: Add the first and third equations to eliminate y.\newline(x−2y−2z)+(2x−2y+2z)=7+8(x - 2y - 2z) + (2x - 2y + 2z) = 7 + 8\newlinex+2x−2y−2y−2z+2z=15x + 2x - 2y - 2y - 2z + 2z = 15\newline3x−4y=153x - 4y = 15
  2. Prepare to eliminate zz: Multiply the second equation by 22 to prepare to eliminate zz with the first equation.\newline2(3x−2y−z)=2(−5)2(3x - 2y - z) = 2(-5)\newline6x−4y−2z=−106x - 4y - 2z = -10
  3. Eliminate z: Add the modified second equation and the third equation to eliminate z.\newline(6x−4y−2z)+(2x−2y+2z)=−10+8(6x - 4y - 2z) + (2x - 2y + 2z) = -10 + 8\newline6x+2x−4y−2y=−26x + 2x - 4y - 2y = -2\newline8x−6y=−28x - 6y = -2
  4. Simplify: Divide the last equation by 22 to simplify.8x−6y=−28x - 6y = -24x−3y=−14x - 3y = -1
  5. Align y terms: Now we have two equations with just x and y:\newline3x−4y=153x - 4y = 15\newline4x−3y=−14x - 3y = -1\newlineMultiply the first equation by 33 and the second by 44 to align y terms.\newline3(3x−4y)=3(15)3(3x - 4y) = 3(15)\newline4(4x−3y)=4(−1)4(4x - 3y) = 4(-1)\newline9x−12y=459x - 12y = 45\newline16x−12y=−416x - 12y = -4
  6. Find xx: Subtract the second equation from the first to find xx.
    (9x−12y)−(16x−12y)=45−(−4)(9x - 12y) - (16x - 12y) = 45 - (-4)
    9x−16x=45+49x - 16x = 45 + 4
    −7x=49-7x = 49
    x=−7x = -7

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