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Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. {:[3x+y=4],[12 x+4y=16]:} Infinitely Many Solutions One Solution No Solutions

Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.\newline3x+y=412x+4y=16 \begin{array}{c} 3 x+y=4 \\ 12 x+4 y=16 \end{array} \newlineInfinitely Many Solutions\newlineOne Solution\newlineNo Solutions

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Q. Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.\newline3x+y=412x+4y=16 \begin{array}{c} 3 x+y=4 \\ 12 x+4 y=16 \end{array} \newlineInfinitely Many Solutions\newlineOne Solution\newlineNo Solutions
  1. Check Coefficients Multiples: We are given the system of equations:\newline3x+y=43x + y = 4\newline12x+4y=1612x + 4y = 16\newlineFirst, we will check if the second equation is a multiple of the first equation.
  2. Check Constant Term: To check if the second equation is a multiple of the first, we can divide the coefficients of the second equation by the corresponding coefficients of the first equation. \newline123=4\frac{12}{3} = 4 and 41=4\frac{4}{1} = 4\newlineSince both coefficients of xx and yy in the second equation are 44 times the corresponding coefficients in the first equation, we can conclude that the second equation is indeed a multiple of the first.
  3. Conclusion: Infinitely Many Solutions: Now, we will check if the constant term of the second equation is also a multiple of the constant term of the first equation.\newline164=4\frac{16}{4} = 4\newlineThis shows that the constant term in the second equation is also 44 times the constant term in the first equation.
  4. Conclusion: Infinitely Many Solutions: Now, we will check if the constant term of the second equation is also a multiple of the constant term of the first equation.\newline164=4\frac{16}{4} = 4\newlineThis shows that the constant term in the second equation is also 44 times the constant term in the first equation.Since all terms in the second equation are multiples of the corresponding terms in the first equation, the two equations represent the same line. Therefore, the system of equations does not have a unique solution. Instead, it has infinitely many solutions, as any point that lies on the line represented by the first equation will also lie on the line represented by the second equation.

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