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Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. {:[-x+2y=5],[2x-4y=-8]:} Infinitely Many Solutions No Solutions One Solution

Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.\newline−x+2y=52x−4y=−8 \begin{array}{l} -x+2 y=5 \\ 2 x-4 y=-8 \end{array} \newlineInfinitely Many Solutions\newlineNo Solutions\newlineOne Solution

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Q. Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.\newline−x+2y=52x−4y=−8 \begin{array}{l} -x+2 y=5 \\ 2 x-4 y=-8 \end{array} \newlineInfinitely Many Solutions\newlineNo Solutions\newlineOne Solution
  1. Given System of Equations: We are given the system of equations:\newline- x+2y=5x + 2y = 5\newline2x−4y=−82x - 4y = -8\newlineFirst, we will try to simplify the second equation by dividing it by 22 to see if it becomes a multiple of the first equation.
  2. Simplify Second Equation: Divide the second equation by 22:\newline(2x−4y)/2=−8/2(2x - 4y) / 2 = -8 / 2\newlinex−2y=−4x - 2y = -4\newlineNow we have the system:\newline−x+2y=5- x + 2y = 5\newlinex−2y=−4x - 2y = -4
  3. Combine Equations: We notice that the second equation is not a multiple of the first equation. However, if we add the two equations together, we should get a new equation that might help us determine the number of solutions.\newline(−x+2y)+(x−2y)=5+(−4)(-x + 2y) + (x - 2y) = 5 + (-4)
  4. Final Result: Perform the addition:\newlinex+x+2y−2y=5−4x + x + 2y - 2y = 5 - 4\newline0=10 = 1\newlineThis is a contradiction because 00 cannot equal 11. This means that the system of equations has no solutions.

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