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Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. {:[x+4y=-5],[4x+16 y=-20]:} Infinitely Many Solutions One Solution No Solutions

Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.\newlinex+4yamp;=−54x+16yamp;=−20 \begin{aligned} x+4 y & =-5 \\ 4 x+16 y & =-20 \end{aligned} \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.\newlinex+4y=−54x+16y=−20 \begin{aligned} x+4 y & =-5 \\ 4 x+16 y & =-20 \end{aligned} \newlineInfinitely Many Solutions\newlineOne Solution\newlineNo Solutions
  1. Given Equations: We are given the system of equations:\newline11. x+4y=−5x + 4y = -5\newline22. 4x+16y=−204x + 16y = -20\newlineLet's first simplify the second equation by dividing all terms by 44 to see if it gives us a clue about the relationship between the two equations.\newline4x4+16y4=−204\frac{4x}{4} + \frac{16y}{4} = \frac{-20}{4}\newlinex+4y=−5x + 4y = -5
  2. Simplify Second Equation: Now we have the simplified system:\newline11. x+4y=−5x + 4y = -5\newline22. x+4y=−5x + 4y = -5\newlineWe can see that both equations are identical, which means that every solution to the first equation is also a solution to the second equation.
  3. Identical Equations: Since both equations represent the same line, there are infinitely many points (x,y)(x, y) that satisfy both equations. Therefore, the system does not have a unique solution, but rather infinitely many solutions.

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