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{:[-20 x+12 y=24],[-5x+3y=6]:} Consider the system of equations. How many (x,y) solutions does this system have? Choose 1 answer: (A) No solutions (B) Exactly one solution (C) Infinitely many solutions (D) None of the above

−20x+12y=24−5x+3y=6 \begin{aligned} -20 x+12 y & =24 \\ -5 x+3 y & =6 \end{aligned} \newlineConsider the system of equations. How many (x,y) (x, y) solutions does this system have?\newlineChoose 11 answer:\newline(A) No solutions\newline(B) Exactly one solution\newline(C) Infinitely many solutions\newline(D) None of the above

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Full solution

Q. −20x+12y=24−5x+3y=6 \begin{aligned} -20 x+12 y & =24 \\ -5 x+3 y & =6 \end{aligned} \newlineConsider the system of equations. How many (x,y) (x, y) solutions does this system have?\newlineChoose 11 answer:\newline(A) No solutions\newline(B) Exactly one solution\newline(C) Infinitely many solutions\newline(D) None of the above
  1. Examine the system of equations: First, let's examine the system of equations:\newline\begin{cases} -20x + 12y = 24,\ -5x + 3y = 6 \end{cases}\newlineWe will check if the second equation is a multiple of the first one.
  2. Check if second equation is a multiple: Divide the first equation by −4 -4 to see if it matches the second equation:\newline(−20x+12y)−4=24−4\frac{(-20x + 12y)}{-4} = \frac{24}{-4}\newline5x−3y=−65x - 3y = -6\newlineThis is the negative of the second equation, which is −5x+3y=6-5x + 3y = 6.
  3. Divide first equation by ext{-}44: Since the second equation is just the negative of the first equation after dividing by ext{-}44, this means that the two equations are actually the same line. Therefore, they have infinitely many solutions because every point on the line is a solution to both equations.

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