Solve the following system of equatians using Gaussian or Gauss-Jardan elimination. \[ \begin{array}{l} 7 x+5 y=-12 \\ 3 x-2 y=12 \end{array} \] Select the correct choice below and, if necessary, fll in the answer bax to complete your chocice. A. There is ane solution. The solution is \( \square \) \( \square \) (Simplify your answer. Type an ondered par) B. There are infinitely mary solutions of the form ( \( x \), \( \square \) ). (Type an expression using \( x \) as the variable ) C. There is no solutian.
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To solve the system of equations using Gaussian elimination, we can write the augmented matrix: \[ \begin{pmatrix} 7 & 5 & | & -12 \\ 3 & -2 & | & 12 \end{pmatrix} \] Next, we want to eliminate \(x\) from the second row. We can do this by multiplying the first row by \(3\) and the second row by \(7\) to make the coefficients of \(x\) the same: \[ 3 \cdot (7x + 5y = -12) \implies 21x + 15y = -36 \] \[ 7 \cdot (3x - 2y = 12) \implies 21x - 14y = 84 \] Now we can form the new augmented matrix: \[ \begin{pmatrix} 21 & 15 & | & -36 \\ 21 & -14 & | & 84 \end{pmatrix} \] Now, we subtract the first row from the second row: \[ 21x - 14y - (21x + 15y) = 84 - (-36) \implies -29y = 120 \] Solving for \(y\): \[ y = \frac{120}{-29} \implies y = -\frac{120}{29} \] Now we substitute \(y\) back into the first equation to find \(x\): \[ 7x + 5\left(-\frac{120}{29}\right) = -12 \] \[ 7x - \frac{600}{29} = -12 \] Multiplying through by \(29\) to eliminate the fraction: \[ 203x - 600 = -348 \] Solving for \(x\): \[ 203x = 252 \implies x = \frac{252}{203} \] So, the solution to the system of equations is: **A. There is one solution. The solution is \(\left( \frac{252}{203}, -\frac{120}{29} \right)\)**.
