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Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. {:[-4x+5y=3],[4x-5y=-6]:} No Solutions One Solution Infinitely Many Solutions

Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.\newline−4x+5yamp;=34x−5yamp;=−6 \begin{aligned} -4 x+5 y & =3 \\ 4 x-5 y & =-6 \end{aligned} \newlineNo Solutions\newlineOne Solution\newlineInfinitely Many Solutions

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Q. Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.\newline−4x+5y=34x−5y=−6 \begin{aligned} -4 x+5 y & =3 \\ 4 x-5 y & =-6 \end{aligned} \newlineNo Solutions\newlineOne Solution\newlineInfinitely Many Solutions
  1. Given Equations: We are given the system of equations:\newline−4x+5y=3-4x + 5y = 3\newline4x−5y=−64x - 5y = -6\newlineFirst, we observe that the coefficients of xx and yy in both equations are the same in magnitude but opposite in sign. This suggests that adding the two equations might eliminate both variables.
  2. Combine Equations: Let's add the two equations:\newline(−4x+5y)+(4x−5y)=3+(−6)(-4x + 5y) + (4x - 5y) = 3 + (-6)\newline−4x+4x+5y−5y=3−6-4x + 4x + 5y - 5y = 3 - 6\newline0x+0y=−30x + 0y = -3\newline0=−30 = -3\newlineThis is a contradiction because 00 cannot equal −3-3. This means that there is no set of values for xx and yy that will satisfy both equations simultaneously.
  3. Identify Contradiction: Since we have arrived at a contradiction, the system of equations has no solutions. The lines represented by these equations are parallel and will never intersect.

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