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Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. {:[x+3y=9],[-2x-6y=-18]:} No Solutions Infinitely Many Solutions One Solution

Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.\newlinex+3y=9−2x−6y=−18 \begin{aligned} x+3 y & =9 \\ -2 x-6 y & =-18 \end{aligned} \newlineNo Solutions\newlineInfinitely Many 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=9−2x−6y=−18 \begin{aligned} x+3 y & =9 \\ -2 x-6 y & =-18 \end{aligned} \newlineNo Solutions\newlineInfinitely Many Solutions\newlineOne Solution
  1. Analyze Equations: Analyze the given system of equations to see if they are multiples of each other.\newlineThe system of equations is:\newline11. x+3y=9x + 3y = 9\newline22. −2x−6y=−18-2x - 6y = -18\newlineWe can multiply the first equation by −2-2 to see if it matches the second equation.\newline−2(x+3y)=−2(9)-2(x + 3y) = -2(9)\newline−2x−6y=−18-2x - 6y = -18
  2. Compare Equations: Compare the resulting equation from Step 11 with the second equation in the system.\newlineAfter multiplying the first equation by −2-2, we get:\newline−2x−6y=−18-2x - 6y = -18\newlineThis is exactly the same as the second equation in the system:\newline−2x−6y=−18-2x - 6y = -18\newlineSince the two equations are identical, this means that every solution to the first equation is also a solution to the second equation.
  3. Conclude Solutions: Conclude the number of solutions based on the comparison.\newlineSince the two equations are identical, the system does not have a unique solution. Instead, it has infinitely many solutions because one equation is a multiple of the other, and they represent the same line.

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