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Solve the system of equations by elimination.\newline2x+y+z=−42x + y + z = -4\newline−x+y−z=4-x + y - z = 4\newline2x+3y+z=−22x + 3y + z = -2

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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−x+y−z=4-x + y - z = 4\newline2x+3y+z=−22x + 3y + z = -2
  1. Combine Equations to Eliminate zz: Combine the first and second equations to eliminate zz.$2x+y+z\$2x + y + z + (-x + y - z) = −4-4 + 44\)2x+y+z−x+y−z=02x + y + z - x + y - z = 0x+2y=0x + 2y = 0
  2. Combine Equations to Simplify: Combine the first and third equations to eliminate zz.$2x+y+z\$2x + y + z + 2x+3y+z2x + 3y + z = −4-4 + (−2-2)\)2x+y+z+2x+3y+z=−62x + y + z + 2x + 3y + z = -64x+4y=−64x + 4y = -6
  3. Divide and Simplify: Divide the equation from the previous step by 44 to simplify.\newline4x+4y=−64x + 4y = -6\newlinex+y=−64x + y = -\frac{6}{4}\newlinex+y=−32x + y = -\frac{3}{2}
  4. Substitute xx into Equation: Substitute x=−2yx = -2y from the equation x+2y=0x + 2y = 0 into x+y=−32x + y = -\frac{3}{2}.\newline(−2y)+y=−32(-2y) + y = -\frac{3}{2}\newline−y=−32-y = -\frac{3}{2}\newliney=32y = \frac{3}{2}
  5. Find xx: Substitute y=32y = \frac{3}{2} into x+2y=0x + 2y = 0 to find xx.
    x+2(32)=0x + 2\left(\frac{3}{2}\right) = 0
    x+3=0x + 3 = 0
    x=−3x = -3
  6. Find zz: Substitute x=−3x = -3 and y=32y = \frac{3}{2} into the first original equation to find zz.
    2(−3)+(32)+z=−42(-3) + \left(\frac{3}{2}\right) + z = -4
    −6+32+z=−4-6 + \frac{3}{2} + z = -4
    −122+32+z=−4-\frac{12}{2} + \frac{3}{2} + z = -4
    −92+z=−4-\frac{9}{2} + z = -4
    z=−4+92z = -4 + \frac{9}{2}
    z=−82+92z = -\frac{8}{2} + \frac{9}{2}
    x=−3x = -300

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