Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. $\newline$ $\begin{array}{r} x+3 y=1 \\ 2 x+6 y=4 \end{array}$ $\newline$ Infinitely Many Solutions $\newline$ No Solutions $\newline$ One Solution

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Algebra 2

Find the number of solutions to a system of equations

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Q. Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.

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\begin{array}{r} x+3 y=1 \\ 2 x+6 y=4 \end{array}

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Infinitely Many Solutions

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No Solutions

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One Solution

Analyze Equations: Analyze the given system of equations. $\newline$ We have the system: $\newline$ $x + 3y = 1$ $\newline$ $2x + 6y = 4$ $\newline$ Let's check if the second equation is a multiple of the first.
Compare Coefficients: Compare the coefficients of the corresponding variables. $\newline$ The first equation has coefficients $1$ for $x$ and $3$ for $y$ . $\newline$ The second equation has coefficients $2$ for $x$ and $6$ for $y$ . $\newline$ Notice that $2$ is twice $1$ , and $6$ is twice $3$ . This suggests that the second equation might be a multiple of the first.
Check Multiples: Check if the second equation is a multiple of the first. $\newline$ If we multiply the first equation by $2$ , we get: $\newline$ $2(x + 3y) = 2(1)$ $\newline$ $2x + 6y = 2$ $\newline$ This is not the same as the second equation given, which is $2x + 6y = 4$ .
Determine Relationship: Determine the relationship between the two equations. $\newline$ Since the second equation is not a multiple of the first, we need to check if they are parallel or if they intersect. $\newline$ If 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.
Check Parallel Lines: Check for parallel lines. $\newline$ For the lines to be parallel, the ratios of the coefficients of $x$ and $y$ should be the same, and the constant terms should be different. $\newline$ The ratios of the coefficients of $x$ and $y$ are the same ( $\frac{1}{3}$ for the first equation and $\frac{2}{6}$ for the second, which simplifies to $\frac{1}{3}$ ), but the constant terms are not multiples of each other ( $1$ is not a multiple of $4$ ).
Conclude Solutions: Conclude the number of solutions. $\newline$ Since 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.