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Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. {:[x+y=-7],[-2x-2y=14]:} Infinitely Many Solutions No Solutions One Solution

Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.\newlinex+y=−7−2x−2y=14 \begin{aligned} x+y & =-7 \\ -2 x-2 y & =14 \end{aligned} \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.\newlinex+y=−7−2x−2y=14 \begin{aligned} x+y & =-7 \\ -2 x-2 y & =14 \end{aligned} \newlineInfinitely Many Solutions\newlineNo Solutions\newlineOne Solution
  1. Analyze System of Equations: Analyze the given system of equations.\newlineThe system of equations is:\newlinex+y=−7x + y = -7\newline−2x−2y=14-2x - 2y = 14\newlineWe will first look for any obvious inconsistencies or proportionalities between the two equations.
  2. Observe Proportional Relationship: Observe that the second equation is −2-2 times the first equation.\newlineIf we multiply the first equation by −2-2, we get:\newline−2(x+y)=−2(−7)-2(x + y) = -2(-7)\newline−2x−2y=14-2x - 2y = 14\newlineThis is exactly the same as the second equation given in the system.
  3. Identify Dependent Equations: Since the second equation is a multiple of the first, the two equations are not independent; they represent the same line.\newlineTherefore, every solution to the first equation is also a solution to the second equation.
  4. Conclude Infinite Solutions: Conclude that the system of equations has infinitely many solutions because both equations represent the same line in the coordinate plane.

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