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Which system has the same solution as the system of equations shown? \[ \begin{array}{l} 8 x+3 y=5 \\ 4 x+2 y=3 \end{array} \] A. \( 8 x+3 y=5 \) \[ -8 x+4 y=6 \] B. \[ \begin{array}{l} 8 x+3 y=5 \\ -8 x-4 y=-6 \end{array} \] C. \( 16 x+6 y=5 \) \( 12 x-6 y=9 \) D. \[ \begin{array}{l} 8 x+3 y=5 \\ 8 x+4 y=3 \end{array} \]

Ask by Peterson Hart. in the United States
Jan 22,2025

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Option B has the same solution as the original system.

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Beyond the Answer

To determine which system has the same solution as the given system, we need to check the linear combinations or multiples of the original equations. First, let's look at option A: The first equation is identical to the original equation. The second, \(-8x + 4y = 6\), can be rewritten by multiplying both sides of the original first equation by -1: \[ -8x - 3y = -5 \implies -8x + 4y = 6 \] (The two equations do not align and thus option A does not have the same solution.) Next, option B contains \(8x + 3y = 5\) again, which is the same as the original equation. The second equation, \(-8x - 4y = -6\), can be manipulated through division by -2 to give us \(4x + 2y = 3\), which is the original second equation. Therefore, option B has the same solution as the system. Now, option C includes a first equation as double the original one: \(16x + 6y = 5\) (this might seem different but isn’t a linear combination) while \(12x - 6y = 9\) does not line up with the original equations’ structure and thus diverges from the original solution. Lastly, in option D, both equations do not represent a linear relationship yielding the same solution. In summary, the answer is option B – it correctly holds the same solution as the system of equations presented.

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