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Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. {:[x+2y=-4],[4x+8y=-16]:} No Solutions One Solution Infinitely Many Solutions

Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.\newlinex+2y=−44x+8y=−16 \begin{array}{c} x+2 y=-4 \\ 4 x+8 y=-16 \end{array} \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.\newlinex+2y=−44x+8y=−16 \begin{array}{c} x+2 y=-4 \\ 4 x+8 y=-16 \end{array} \newlineNo Solutions\newlineOne Solution\newlineInfinitely Many Solutions
  1. Check Equation Multiples: We are given the system of equations:\newline11. x+2y=−4x + 2y = -4\newline22. 4x+8y=−164x + 8y = -16\newlineWe will first check if the second equation is a multiple of the first equation.
  2. Divide Second Equation: Divide the second equation by 44 to see if it matches the first equation:\newline(4x+8y)/4=−16/4(4x + 8y) / 4 = -16 / 4\newlineThis simplifies to:\newlinex+2y=−4x + 2y = -4
  3. Identical Equations: We observe that the second equation, after dividing by 44, is identical to the first equation. This means that the two equations represent the same line.
  4. Infinite Solutions: Since both equations represent the same line, every point on the line is a solution to the system. Therefore, the system has infinitely many solutions.

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