Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. $\newline$ $\begin{array}{c} 3 x+5 y=-6 \\ 6 x+10 y=-12 \end{array}$ $\newline$ Infinitely Many Solutions $\newline$ No Solutions $\newline$ One Solution

Full solution

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

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\begin{array}{c} 3 x+5 y=-6 \\ 6 x+10 y=-12 \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 system of equations. $\newline$ We have the system: $\newline$ $3x + 5y = -6$ $\newline$ $6x + 10y = -12$ $\newline$ We will check if the second equation is a multiple of the first one.
Compare Coefficients: Compare the coefficients of the corresponding variables and the constants. $\newline$ The second equation has coefficients that are exactly twice the coefficients of the first equation ( $6x$ is $2$ times $3x$ , $10y$ is $2$ times $5y$ , and $-12$ is $2$ times $-6$ ). $\newline$ This suggests that the second equation might be a multiple of the first.
Determine Multiplicity: Determine if the second equation is a multiple of the first. $\newline$ Divide the second equation by $2$ : $\newline$ $(6x + 10y = -12) / 2$ $\newline$ $3x + 5y = -6$ $\newline$ This is the same as the first equation.
Conclude Solution: Conclude the type of solution the system has. $\newline$ Since the second equation is a multiple of the first, the two equations are essentially the same line. Therefore, every solution to the first equation is also a solution to the second equation. $\newline$ This means the system has infinitely many solutions.