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Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. {:[x+3y=1],[2x+6y=4]:} 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+3y=12x+6y=4 \begin{array}{r} x+3 y=1 \\ 2 x+6 y=4 \end{array} \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+3y=12x+6y=4 \begin{array}{r} x+3 y=1 \\ 2 x+6 y=4 \end{array} \newlineInfinitely Many Solutions\newlineNo Solutions\newlineOne Solution
  1. Analyze Equations: Analyze the given system of equations.\newlineWe have the system:\newlinex+3y=1x + 3y = 1\newline2x+6y=42x + 6y = 4\newlineLet's check if the second equation is a multiple of the first.
  2. Compare Coefficients: Compare the coefficients of the corresponding variables.\newlineThe first equation has coefficients 11 for xx and 33 for yy.\newlineThe second equation has coefficients 22 for xx and 66 for yy.\newlineNotice that 22 is twice 11, and 66 is twice 33. This suggests that the second equation might be a multiple of the first.
  3. Check Multiples: Check if the second equation is a multiple of the first.\newlineIf we multiply the first equation by 22, we get:\newline2(x+3y)=2(1)2(x + 3y) = 2(1)\newline2x+6y=22x + 6y = 2\newlineThis is not the same as the second equation given, which is 2x+6y=42x + 6y = 4.
  4. Determine Relationship: Determine the relationship between the two equations.\newlineSince the second equation is not a multiple of the first, we need to check if they are parallel or if they intersect.\newlineIf the equations were identical after simplification, they would have infinitely many solutions. If they were parallel and different, they would have no solutions. If they intersect at a point, they would have one solution.
  5. Check Parallel Lines: Check for parallel lines.\newlineFor the lines to be parallel, the ratios of the coefficients of xx and yy should be the same, and the constant terms should be different.\newlineThe ratios of the coefficients of xx and yy are the same (13\frac{1}{3} for the first equation and 26\frac{2}{6} for the second, which simplifies to 13\frac{1}{3}), but the constant terms are not multiples of each other (11 is not a multiple of 44).
  6. Conclude Solutions: Conclude the number of solutions.\newlineSince the ratios of the coefficients are the same but the constant terms are not multiples of each other, the lines are parallel and distinct. Therefore, the system of equations has no solutions.

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