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Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. {:[2x+3y=4],[4x+6y=8]:} No Solutions One Solution Infinitely Many Solutions

Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.\newline2x+3y=44x+6y=8 \begin{array}{l} 2 x+3 y=4 \\ 4 x+6 y=8 \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.\newline2x+3y=44x+6y=8 \begin{array}{l} 2 x+3 y=4 \\ 4 x+6 y=8 \end{array} \newlineNo Solutions\newlineOne Solution\newlineInfinitely Many Solutions
  1. Analyze System of Equations: Analyze the given system of equations to see if they are multiples of each other.\newlineThe system of equations is:\newline2x+3y=42x + 3y = 4\newline4x+6y=84x + 6y = 8\newlineWe can see that the second equation is exactly twice the first equation.
  2. Divide Second Equation: Divide the second equation by 22 to check if it becomes identical to the first equation.\newlineegin{equation}(44x + 66y = 88) \div 22\end{equation} gives us 2x+3y=42x + 3y = 4, which is the same as the first equation.
  3. Identical Equations Conclusion: Since both equations are identical after simplification, this means that every solution to the first equation is also a solution to the second equation.\newlineTherefore, the system of equations does not have a unique solution. Instead, it has infinitely many solutions because one equation is a multiple of the other.

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