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How many solutions does the system of equations below have?\newline−3x+y−3z=4-3x + y - 3z = 4\newlinex−y−z=16x - y - z = 16\newline3x−y+3z=−43x - y + 3z = -4\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−3x+y−3z=4-3x + y - 3z = 4\newlinex−y−z=16x - y - z = 16\newline3x−y+3z=−43x - y + 3z = -4\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions
  1. Combine Equations: Combine the first and third equations to eliminate yy and zz.(−3x+y−3z)+(3x−y+3z)=4+(−4)(-3x + y - 3z) + (3x - y + 3z) = 4 + (-4)0x+0y+0z=00x + 0y + 0z = 0This simplifies to 0=00 = 0, which is always true.
  2. Combine Second and Third: Now, let's combine the second and third equations.\newline(x−y−z)+(3x−y+3z)=16+(−4)(x - y - z) + (3x - y + 3z) = 16 + (-4)\newline4x−2y+2z=124x - 2y + 2z = 12\newlineDivide the entire equation by 22 to simplify.\newline2x−y+z=62x - y + z = 6
  3. Simplified Equations: Look at the simplified equations:\newline0=00 = 0 (from combining the first and third equations)\newline2x−y+z=62x - y + z = 6 (from combining the second and third equations)\newlineThe first equation is always true, and the second equation represents a plane in three-dimensional space.
  4. Infinite Solutions: Since the first equation is always true, it doesn't restrict the solutions. The second equation represents a plane, and there's no third independent equation to further restrict the solutions.\newlineTherefore, the system has infinitely many solutions because any point on the plane represented by the second equation will satisfy the system.

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