Solve the system by the addition method. \[ \begin{array}{r}4 x+12 y=8 \\ 3 x-12 y=6\end{array} \] Aelect the correct choice below and, if necessary, fill in the answer box to complete your choice. B. There are infinitely many solutions. C. There is no solution.
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Bonus Knowledge
To solve the system of equations using the addition method, we first align the equations: 1. \( 4x + 12y = 8 \) 2. \( 3x - 12y = 6 \) Next, we can add these equations together. Notice that the \(12y\) and \(-12y\) will cancel out: \[ (4x + 12y) + (3x - 12y) = 8 + 6 \] This simplifies to: \[ 7x = 14 \] Now, we can solve for \(x\): \[ x = 2 \] Now we can substitute \(x = 2\) back into one of the original equations to find \(y\). Using the first equation: \[ 4(2) + 12y = 8 \] Which simplifies to: \[ 8 + 12y = 8 \] \[ 12y = 0 \Rightarrow y = 0 \] Thus, the solution to this system of equations is \(x = 2\) and \(y = 0\), so we have a unique solution: \((2, 0)\). Since we found a solution, the correct choice is that there is one unique solution. Hence, the answer is: A. There is a unique solution: \( (2, 0) \).
