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Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. {:[2x+5y=-7],[-2x-2y=6]:} One Solution No Solutions Infinitely Many Solutions

Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.\newline2x+5y=−7−2x−2y=6 \begin{aligned} 2 x+5 y & =-7 \\ -2 x-2 y & =6 \end{aligned} \newlineOne Solution\newlineNo Solutions\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.\newline2x+5y=−7−2x−2y=6 \begin{aligned} 2 x+5 y & =-7 \\ -2 x-2 y & =6 \end{aligned} \newlineOne Solution\newlineNo Solutions\newlineInfinitely Many Solutions
  1. Combine and Simplify Equations: We have the system of equations:\newline2x+5y=−72x + 5y = -7\newline−2x−2y=6-2x - 2y = 6\newlineFirst, let's add the two equations together to see if we can simplify the system.\newline(2x+5y)+(−2x−2y)=−7+6(2x + 5y) + (-2x - 2y) = -7 + 6\newline2x−2x+5y−2y=−12x - 2x + 5y - 2y = -1\newline0x+3y=−10x + 3y = -1\newline3y=−13y = -1\newlineWe can simplify this to:\newliney=−13y = -\frac{1}{3}
  2. Substitute yy to Find xx: Now that we have the value of yy, we can substitute it back into one of the original equations to find the value of xx. Let's use the first equation:\newline2x+5(−13)=−72x + 5(-\frac{1}{3}) = -7\newline2x−53=−72x - \frac{5}{3} = -7\newlineMultiply both sides by 33 to clear the fraction:\newline6x−5=−216x - 5 = -21\newlineAdd 55 to both sides:\newline6x=−166x = -16\newlineDivide by xx00:\newlinexx11\newlinexx22
  3. Check Solution: We have found specific values for xx and yy, which means the system of equations has exactly one solution. To ensure there are no mistakes, we should check these values in the second equation as well.\newline−2(−83)−2(−13)=6-2(-\frac{8}{3}) - 2(-\frac{1}{3}) = 6\newline163+23=6\frac{16}{3} + \frac{2}{3} = 6\newline183=6\frac{18}{3} = 6\newline6=66 = 6\newlineThe values satisfy the second equation as well, confirming our solution is correct.

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