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How many solutions does the system of equations below have?\newline−x−2y−z=14-x - 2y - z = 14\newline−2x+y+3z=18-2x + y + 3z = 18\newline2x−y−3z=−172x - y - 3z = -17\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−x−2y−z=14-x - 2y - z = 14\newline−2x+y+3z=18-2x + y + 3z = 18\newline2x−y−3z=−172x - y - 3z = -17\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:\newline11) −x−2y−z=14-x - 2y - z = 14\newline22) −2x+y+3z=18-2x + y + 3z = 18\newline33) 2x−y−3z=−172x - y - 3z = -17
  2. Elimination Method: We can try to solve the system using the elimination or substitution method. Let's start by adding equations 2)2) and 3)3) to eliminate yy and zz.(−2x+y+3z)+(2x−y−3z)=18+(−17)(-2x + y + 3z) + (2x - y - 3z) = 18 + (-17)
  3. Combine Equations: Simplifying the left side, we get: −2x+2x+y−y+3z−3z=0-2x + 2x + y - y + 3z - 3z = 0
  4. Simplify Left Side: Simplifying the right side, we get: 18−17=118 - 17 = 1
  5. Simplify Right Side: But on the left side, all the variables cancel out, so we are left with:\newline0=10 = 1\newlineThis is a contradiction, which means there is no solution to the system of equations.

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