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Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. {:[-5x+3y=2],[5x-3y=-5]:} 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−5x+3y=25x−3y=−5 \begin{aligned} -5 x+3 y & =2 \\ 5 x-3 y & =-5 \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−5x+3y=25x−3y=−5 \begin{aligned} -5 x+3 y & =2 \\ 5 x-3 y & =-5 \end{aligned} \newlineNo Solutions\newlineOne Solution\newlineInfinitely Many Solutions
  1. Write Equations: Write down the system of equations.\newline−5x+3y=2-5x + 3y = 2\newline5x−3y=−55x - 3y = -5
  2. Add Equations: Add the two equations together to see if they are consistent or inconsistent.\newline(−5x+3y)+(5x−3y)=2+(−5)(-5x + 3y) + (5x - 3y) = 2 + (-5)\newline−5x+3y+5x−3y=−3-5x + 3y + 5x - 3y = -3\newline0=−30 = -3
  3. Check Consistency: Since adding the left sides of the equations results in 00 and the right sides do not add up to 00, the equations are inconsistent.\newlineThis means that there is no set of values for xx and yy that will satisfy both equations simultaneously.
  4. Conclude No Solutions: Conclude that the system of equations has no solutions because the equations are inconsistent.

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