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How many solutions does the system have? {[20 x-5y=5],[4x-y=1]:} Choose 1 answer: (A) Exactly one solution (B) No solutions (c) Infinitely many solutions

How many solutions does the system have?\newline{20x−5y=54x−y=1 \left\{\begin{array}{l} 20 x-5 y=5 \\ 4 x-y=1 \end{array}\right. \newlineChoose 11 answer:\newline(A) Exactly one solution\newline(B) No solutions\newline(C) Infinitely many solutions

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Q. How many solutions does the system have?\newline{20x−5y=54x−y=1 \left\{\begin{array}{l} 20 x-5 y=5 \\ 4 x-y=1 \end{array}\right. \newlineChoose 11 answer:\newline(A) Exactly one solution\newline(B) No solutions\newline(C) Infinitely many solutions
  1. Write Equations: Write down the system of equations to analyze.\newlineThe system is:\newline{20x−5y=54x−y=1 \begin{cases} 20x - 5y = 5 \\ 4x - y = 1 \end{cases}
  2. Compare Equations: Look for a quick way to compare the two equations. Notice that the second equation can be multiplied by 55 to see if it becomes similar to the first equation.\newlineMultiplying the second equation by 55 gives us:\newline5(4x−y)=5(1) 5(4x - y) = 5(1) \newline20x−5y=5 20x - 5y = 5
  3. Identify Same Line: Compare the new form of the second equation with the first equation.\newlineAfter multiplying the second equation by 55, we see that it becomes identical to the first equation:\newline20x−5y=5 20x - 5y = 5 \newlineThis means that the two equations are actually the same line.
  4. Conclude Solutions: Conclude the number of solutions for the system.\newlineSince both equations represent the same line, the system does not have a unique solution. Instead, it has infinitely many solutions because every point on the line is a solution to the system.

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