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

Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.\newlinex+3yamp;=22x+6yamp;=4 \begin{aligned} x+3 y & =2 \\ 2 x+6 y & =4 \end{aligned} \newlineOne Solution\newlineInfinitely Many Solutions\newlineNo Solutions

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Q. Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.\newlinex+3y=22x+6y=4 \begin{aligned} x+3 y & =2 \\ 2 x+6 y & =4 \end{aligned} \newlineOne Solution\newlineInfinitely Many Solutions\newlineNo Solutions
  1. Given Equations: We are given the system of equations:\newlinex+3y=2x + 3y = 2\newline2x+6y=42x + 6y = 4\newlineFirst, we observe that the second equation is just the first equation multiplied by 22. This suggests that the two equations may actually be the same line, which would mean they have infinitely many solutions in common. However, we need to verify this by simplifying the equations further.
  2. Observation: Let's divide the second equation by 22 to see if it becomes identical to the first equation:\newline(2x+6y)/2=4/2(2x + 6y) / 2 = 4 / 2\newlinex+3y=2x + 3y = 2\newlineNow we can see that the second equation simplifies to the first equation, confirming that they are indeed the same line.
  3. Verification: Since both equations represent the same line, every point on that line is a solution to the system. Therefore, the system does not have a unique solution or no solution; instead, it has infinitely many solutions.

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