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How many solutions does the system of equations below have?\newline−2x+3y−3z=−13-2x + 3y - 3z = -13\newline−2x+2y−3z=−18-2x + 2y - 3z = -18\newline2x+y−2z=32x + y - 2z = 3\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+3y−3z=−13-2x + 3y - 3z = -13\newline−2x+2y−3z=−18-2x + 2y - 3z = -18\newline2x+y−2z=32x + y - 2z = 3\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions
  1. Combine Equations: Combine the first two equations to eliminate xx.\(\newline\)(−2x+3y−3z-2x + 3y - 3z) - (−2x+2y−3z-2x + 2y - 3z) = −13-13 - (−18-18)\(\newline\)−2x+3y−3z+2x−2y+3z=−13+18-2x + 3y - 3z + 2x - 2y + 3z = -13 + 18\(\newline\)y=5y = 5
  2. Substitute yy into third: Substitute y=5y = 5 into the third equation.2x+y−2z=32x + y - 2z = 32x+5−2z=32x + 5 - 2z = 32x−2z=−22x - 2z = -2
  3. Simplify by Division: Divide the last equation by 22 to simplify.\newline2x−2z2=−22\frac{2x - 2z}{2} = \frac{-2}{2}\newlinex−z=−1x - z = -1
  4. Equations with x and z: Now we have two equations with x and z.\newlinex−z=−1x - z = -1\newline−2x+3y−3z=−13-2x + 3y - 3z = -13\newlineSubstitute y=5y = 5 into the second equation.\newline−2x+3(5)−3z=−13-2x + 3(5) - 3z = -13\newline−2x+15−3z=−13-2x + 15 - 3z = -13\newline−2x−3z=−28-2x - 3z = -28

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