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Solve the system of equations by elimination.\newline−2x−3y+3z=−19-2x - 3y + 3z = -19\newline3x+y+2z=−163x + y + 2z = -16\newline2x−y+2z=−102x - y + 2z = -10

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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.\newline−2x−3y+3z=−19-2x - 3y + 3z = -19\newline3x+y+2z=−163x + y + 2z = -16\newline2x−y+2z=−102x - y + 2z = -10
  1. Add Equations to Eliminate y: First, let's add the second and third equations to eliminate y.\newline(3x+y+2z)+(2x−y+2z)=−16+(−10)(3x + y + 2z) + (2x - y + 2z) = -16 + (-10)\newline5x+4z=−265x + 4z = -26
  2. Multiply Equations to Eliminate y: Now, let's multiply the first equation by 33 and the second equation by 22 so we can eliminate y again.\newline(−2x−3y+3z)×3=−19×3(-2x - 3y + 3z) \times 3 = -19 \times 3\newline(3x+y+2z)×2=−16×2(3x + y + 2z) \times 2 = -16 \times 2\newline−6x−9y+9z=−57-6x - 9y + 9z = -57\newline6x+2y+4z=−326x + 2y + 4z = -32
  3. Add Equations to Eliminate y: Next, we add the new equations from the previous step to eliminate y.\newline(−6x−9y+9z)+(6x+2y+4z)=−57+(−32)(-6x - 9y + 9z) + (6x + 2y + 4z) = -57 + (-32)\newline−7y+13z=−89-7y + 13z = -89
  4. Align Coefficients of zz: Now we have two equations:\newline5x+4z=−265x + 4z = -26\newline−7y+13z=−89-7y + 13z = -89\newlineWe need to find one more equation to eliminate another variable.
  5. Add Equations to Eliminate z: Let's multiply the first original equation by 22 and the third original equation by 33 to align the coefficients of z.(−2x−3y+3z)×2=−19×2(-2x - 3y + 3z) \times 2 = -19 \times 2(2x−y+2z)×3=−10×3(2x - y + 2z) \times 3 = -10 \times 3−4x−6y+6z=−38-4x - 6y + 6z = -386x−3y+6z=−306x - 3y + 6z = -30
  6. Solve for x: Now, we add these two new equations to eliminate z.\newline(−4x−6y+6z)+(6x−3y+6z)=−38+(−30)(-4x - 6y + 6z) + (6x - 3y + 6z) = -38 + (-30)\newline2x−9y=−682x - 9y = -68
  7. Substitute xx into Equation: We now have three equations with two variables each: 5x+4z=−265x + 4z = -26 −7y+13z=−89-7y + 13z = -89 2x−9y=−682x - 9y = -68 We can solve one of these equations for a variable and substitute it into another equation.
  8. Multiply to Remove Fraction: Let's solve the third equation for xx.2x−9y=−682x - 9y = -682x=9y−682x = 9y - 68x=9y−682x = \frac{9y - 68}{2}
  9. Add Equations to Eliminate yy: Substitute xx into the first equation we found, 5x+4z=−265x + 4z = -26.\newline5(9y−682)+4z=−265\left(\frac{9y - 68}{2}\right) + 4z = -26\newline45y−3402+4z=−26\frac{45y - 340}{2} + 4z = -26
  10. Add Equations to Eliminate yy: Substitute xx into the first equation we found, 5x+4z=−265x + 4z = -26.
    5(9y−682)+4z=−265\left(\frac{9y - 68}{2}\right) + 4z = -26
    45y−3402+4z=−26\frac{45y - 340}{2} + 4z = -26Multiply everything by 22 to get rid of the fraction.
    45y−340+8z=−5245y - 340 + 8z = -52
  11. Add Equations to Eliminate yy: Substitute xx into the first equation we found, 5x+4z=−265x + 4z = -26.
    5(9y−682)+4z=−265\left(\frac{9y - 68}{2}\right) + 4z = -26
    45y−3402+4z=−26\frac{45y - 340}{2} + 4z = -26Multiply everything by 22 to get rid of the fraction.
    45y−340+8z=−5245y - 340 + 8z = -52Now, let's add this equation to the second equation we found, −7y+13z=−89-7y + 13z = -89, to eliminate yy.
    (45y−340+8z)+(−7y+13z)=−52+(−89)(45y - 340 + 8z) + (-7y + 13z) = -52 + (-89)
    xx00

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