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How many solutions does the system of equations below have?\newline−x+2y+3z=−10-x + 2y + 3z = -10\newline−3x−2y−2z=−4-3x - 2y - 2z = -4\newline−3x+2y−z=10-3x + 2y - z = 10\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions

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Q. How many solutions does the system of equations below have?\newline−x+2y+3z=−10-x + 2y + 3z = -10\newline−3x−2y−2z=−4-3x - 2y - 2z = -4\newline−3x+2y−z=10-3x + 2y - z = 10\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions
  1. Write Equations: First, let's write down the system of equations to see if we can spot anything obvious.\newline−x+2y+3z=−10-x + 2y + 3z = -10\newline−3x−2y−2z=−4-3x - 2y - 2z = -4\newline−3x+2y−z=10-3x + 2y - z = 10
  2. Eliminate Variables: We can try to add the first and second equations to eliminate yy.(−x+2y+3z)+(−3x−2y−2z)=−10+(−4)(-x + 2y + 3z) + (-3x - 2y - 2z) = -10 + (-4)This simplifies to −4x+z=−14-4x + z = -14
  3. Simplify Equations: Now, let's add the second and third equations to eliminate yy again.(−3x−2y−2z)+(−3x+2y−z)=−4+10(-3x - 2y - 2z) + (-3x + 2y - z) = -4 + 10This simplifies to −6x−3z=6-6x - 3z = 6
  4. Match Coefficients: We can multiply the equation from step 33 by 33 to match the zz coefficient in the equation from step 44.\newline(−4x+z)×3=−14×3(-4x + z) \times 3 = -14 \times 3\newlineThis gives us −12x+3z=−42-12x + 3z = -42
  5. Eliminate Variable zz: Now we can add the equation from step 44 and the new equation from step 55 to eliminate zz.(−12x+3z)+(−6x−3z)=−42+6(-12x + 3z) + (-6x - 3z) = -42 + 6This simplifies to −18x=−36-18x = -36
  6. Solve for x: Divide both sides by −18-18 to solve for x.\newline−18x/−18=−36/−18-18x / -18 = -36 / -18\newlineThis gives us x=2x = 2
  7. Substitute xx into Equation: Now that we have a value for xx, we can substitute it back into the equations to solve for yy and zz. Let's substitute x=2x = 2 into the equation from step 33. −4(2)+z=−14-4(2) + z = -14 This gives us −8+z=−14-8 + z = -14

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