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How many solutions does the system of equations below have?\newlinex+3y−z=−5x + 3y - z = -5\newline−x−3y+z=5-x - 3y + z = 5\newline2x+3y+z=172x + 3y + z = 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?\newlinex+3y−z=−5x + 3y - z = -5\newline−x−3y+z=5-x - 3y + z = 5\newline2x+3y+z=172x + 3y + z = 17\newlineChoices:\newline(A) no solution\newline(B) one solution\newline(C) infinitely many solutions
  1. Add Equations: Add the first and second equations to eliminate xx, yy, and zz.\newline(x+3y−z)+(−x−3y+z)=−5+5(x + 3y - z) + (-x - 3y + z) = -5 + 5\newline0=00 = 0
  2. Check Dependence: Since 0=00 = 0 is a true statement, it means the first two equations are dependent, and they represent the same plane.
  3. Check Consistency: Now, check if the third equation is consistent with the first two by adding the first equation to the third.\newline(x+3y−z)+(2x+3y+z)=−5+17(x + 3y - z) + (2x + 3y + z) = -5 + 17\newline3x+6y=123x + 6y = 12
  4. Simplify Equation: Divide the entire equation by 33 to simplify.\newlinex+2y=4x + 2y = 4
  5. Different Plane: This new equation is not a multiple of the first equation, which means the third equation represents a different plane that intersects the plane represented by the first two equations at a line.
  6. Infinite Solutions: Since all three planes intersect at a line, the system has infinitely many solutions.

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