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Solve the system of equations by elimination.\newline3x−y−z=73x - y - z = 7\newline−2x+y+3z=−7-2x + y + 3z = -7\newline2x−2y+z=142x - 2y + z = 14

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Q. Solve the system of equations by elimination.\newline3x−y−z=73x - y - z = 7\newline−2x+y+3z=−7-2x + y + 3z = -7\newline2x−2y+z=142x - 2y + z = 14
  1. Eliminate y by adding equations: First, let's add the first and second equations to eliminate y.\newline(3x−y−z)+(−2x+y+3z)=7+(−7)(3x - y - z) + (-2x + y + 3z) = 7 + (-7)\newline3x−y−z−2x+y+3z=03x - y - z - 2x + y + 3z = 0\newlinex+2z=0x + 2z = 0
  2. Simplify third equation: Now, let's multiply the third equation by 12\frac{1}{2} to simplify it.\newline(12)(2x−2y+z)=(12)(14)\left(\frac{1}{2}\right)(2x - 2y + z) = \left(\frac{1}{2}\right)(14)\newlinex−y+(12)z=7x - y + \left(\frac{1}{2}\right)z = 7
  3. Eliminate y by adding equations: Next, we'll add the modified third equation to the first equation to eliminate y. \newline(x−y+(1/2)z)+(3x−y−z)=7+7(x - y + (1/2)z) + (3x - y - z) = 7 + 7\newlinex−y+(1/2)z+3x−y−z=14x - y + (1/2)z + 3x - y - z = 14\newline4x−2y−(1/2)z=144x - 2y - (1/2)z = 14
  4. Eliminate z by adding equations: We can now add the new equation from the previous step to the modified second equation to eliminate z.\newline(4x−2y−12z)+(x+2z)=14+0(4x - 2y - \frac{1}{2}z) + (x + 2z) = 14 + 0\newline4x−2y−12z+x+2z=144x - 2y - \frac{1}{2}z + x + 2z = 14\newline5x−2y+32z=145x - 2y + \frac{3}{2}z = 14
  5. Get rid of fraction: Let's multiply the new equation by 22 to get rid of the fraction.\newline2(5x−2y+32z)=2(14)2(5x - 2y + \frac{3}{2}z) = 2(14)\newline10x−4y+3z=2810x - 4y + 3z = 28
  6. Eliminate y by adding equations: Now, we'll add the modified third equation to the new equation to eliminate y.\newline(10x−4y+3z)+(x−y+12z)=28+7(10x - 4y + 3z) + (x - y + \frac{1}{2}z) = 28 + 7\newline10x−4y+3z+x−y+12z=3510x - 4y + 3z + x - y + \frac{1}{2}z = 35\newline11x−5y+72z=3511x - 5y + \frac{7}{2}z = 35
  7. Solve for x: We can now solve for x by adding the first and third equations.\newline(3x−y−z)+(11x−5y+72z)=7+35(3x - y - z) + (11x - 5y + \frac{7}{2}z) = 7 + 35\newline3x−y−z+11x−5y+72z=423x - y - z + 11x - 5y + \frac{7}{2}z = 42\newline14x−6y+52z=4214x - 6y + \frac{5}{2}z = 42
  8. Divide to solve for x: Divide the new equation by 1414 to solve for xx.
    (14x−6y+(5/2)z)/14=42/14(14x - 6y + (5/2)z) / 14 = 42 / 14
    x−(6/14)y+(5/28)z=3x - (6/14)y + (5/28)z = 3
    x=3+(6/14)y−(5/28)zx = 3 + (6/14)y - (5/28)z

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