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How many solutions does the system of equations below have?\newline−2x−y−3z=−13-2x - y - 3z = -13\newline−x−y+z=−3-x - y + z = -3\newline3x+2y+2z=163x + 2y + 2z = 16\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions

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Determine the number of solutions to a system of equations in three variablesright chevron icon

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Q. How many solutions does the system of equations below have?\newline−2x−y−3z=−13-2x - y - 3z = -13\newline−x−y+z=−3-x - y + z = -3\newline3x+2y+2z=163x + 2y + 2z = 16\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions
  1. Combine equations to eliminate yy: First, let's add the first and second equations to eliminate yy.(−2x−y−3z)+(−x−y+z)=−13+(−3)(-2x - y - 3z) + (-x - y + z) = -13 + (-3)−2x−x−y−y−3z+z=−16-2x - x - y - y - 3z + z = -16−3x−2y−2z=−16-3x - 2y - 2z = -16
  2. Use multiplication to eliminate y again: Now, let's multiply the second equation by 22 and add it to the third equation to eliminate y again.\newline2(−x−y+z)+(3x+2y+2z)=2(−3)+162(-x - y + z) + (3x + 2y + 2z) = 2(-3) + 16\newline−2x−2y+2z+3x+2y+2z=−6+16-2x - 2y + 2z + 3x + 2y + 2z = -6 + 16\newlinex+4z=10x + 4z = 10
  3. Form new system of equations: We have the new system of equations:\newline−3x−2y−2z=−16-3x - 2y - 2z = -16\newlinex+4z=10x + 4z = 10\newline3x+2y+2z=163x + 2y + 2z = 16
  4. Find relationship between x and z: Let's multiply the second equation by 33 and add it to the first equation to find a relationship between x and z.\newline3(x+4z)+(−3x−2y−2z)=3(10)+(−16)3(x + 4z) + (-3x - 2y - 2z) = 3(10) + (-16)\newline3x+12z−3x−2y−2z=30−163x + 12z - 3x - 2y - 2z = 30 - 16\newline10z−2y=1410z - 2y = 14
  5. Solve for y in terms of z: Now, let's solve for y in terms of z using the equation 10z−2y=1410z - 2y = 14. \newline2y=10z−142y = 10z - 14\newliney=5z−7y = 5z - 7
  6. Substitute yy into third equation: Substitute y=5z−7y = 5z - 7 into the third equation 3x+2y+2z=163x + 2y + 2z = 16.
    3x+2(5z−7)+2z=163x + 2(5z - 7) + 2z = 16
    3x+10z−14+2z=163x + 10z - 14 + 2z = 16
    3x+12z=303x + 12z = 30
  7. Solve for x in terms of z: Now, let's solve for x in terms of z using the equation 3x+12z=303x + 12z = 30.\newline3x=30−12z3x = 30 - 12z\newlinex=10−4zx = 10 - 4z
  8. Express xx and yy in terms of zz: We have expressions for xx and yy in terms of zz:
    x=10−4zx = 10 - 4z
    y=5z−7y = 5z - 7
    This means we can choose any value for zz and find corresponding values for xx and yy, indicating that there are infinitely many solutions.

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