Sun Feb 2 Dicom Suppose the following system of equations has a solution of \( (7,4) \), where \( A, B, C, D, E \), and \( F \) are real numbers. \[ \begin{array}{l} A x+B y=C \\ D x+E y=F \end{array} \] Which systems are also guaranteed to have a solution of \( (7,4) \) ? Select all that apply. \[ \begin{array}{l} \left(A+\frac{1}{4} D\right) x+\left(B+\frac{1}{4} E\right) y=C+\frac{1}{4} F \\ D x+E y=F \end{array} \] \[ \begin{array}{l} \left(A+\frac{2}{3} E\right) x+\left(B+\frac{2}{3} D\right) y=C+\frac{2}{3} F \\ D x+E y=F \end{array} \] \[ \begin{array}{l} A x+B y=C \\ (3 A+D) x+(3 B+E) y=3 C+F \end{array} \] \[ \begin{array}{l} A x+B y=C \\ (-2 A+D) x+(2 B-E) y=-2 C+F \end{array} \] \[ \begin{array}{l} A x+B y=C \\ \left(D-\frac{1}{5} A\right) x+\left(E-\frac{1}{5} B\right) y=F-\frac{1}{5} C \end{array} \]

To determine which systems are guaranteed to have the solution \((7, 4)\), let's analyze each option based on the original system: \[ \begin{cases} A x + B y = C \\ D x + E y = F \end{cases} \] Given that \((7, 4)\) is a solution, we have: \[ \begin{cases} 7A + 4B = C \\ 7D + 4E = F \end{cases} \] Now, let's evaluate each of the provided systems: 1. **First System:** \[ \begin{cases} \left(A + \frac{1}{4} D\right) x + \left(B + \frac{1}{4} E\right) y = C + \frac{1}{4} F \\ D x + E y = F \end{cases} \] **Verification:** \[ \left(A + \frac{1}{4} D\right) \cdot 7 + \left(B + \frac{1}{4} E\right) \cdot 4 = 7A + 4B + \frac{1}{4}(7D + 4E) = C + \frac{1}{4} F \] Since \(7D + 4E = F\), the equation holds true. **Conclusion:** \(\boxed{\text{This system is guaranteed to have }(7,4)}\) 2. **Second System:** \[ \begin{cases} \left(A + \frac{2}{3} E\right) x + \left(B + \frac{2}{3} D\right) y = C + \frac{2}{3} F \\ D x + E y = F \end{cases} \] **Verification:** \[ \left(A + \frac{2}{3} E\right) \cdot 7 + \left(B + \frac{2}{3} D\right) \cdot 4 \neq C + \frac{2}{3} F \quad \text{(in general)} \] This equality only holds under specific conditions, which are not guaranteed by the original system. **Conclusion:** \(\boxed{\text{This system is not necessarily guaranteed to have }(7,4)}\) 3. **Third System:** \[ \begin{cases} A x + B y = C \\ (3A + D) x + (3B + E) y = 3C + F \end{cases} \] **Verification:** \[ (3A + D) \cdot 7 + (3B + E) \cdot 4 = 3(7A + 4B) + (7D + 4E) = 3C + F \] This equality holds true. **Conclusion:** \(\boxed{\text{This system is guaranteed to have }(7,4)}\) 4. **Fourth System:** \[ \begin{cases} A x + B y = C \\ (-2A + D) x + (2B - E) y = -2C + F \end{cases} \] **Verification:** \[ (-2A + D) \cdot 7 + (2B - E) \cdot 4 = -2(7A + 4B) + (7D + 4E) = -2C + F \] This equality holds true. **Conclusion:** \(\boxed{\text{This system is guaranteed to have }(7,4)}\) 5. **Fifth System:** \[ \begin{cases} A x + B y = C \\ \left(D - \frac{1}{5} A\right) x + \left(E - \frac{1}{5} B\right) y = F - \frac{1}{5} C \end{cases} \] **Verification:** \[ \left(D - \frac{1}{5} A\right) \cdot 7 + \left(E - \frac{1}{5} B\right) \cdot 4 = 7D + 4E - \frac{1}{5}(7A + 4B) = F - \frac{1}{5} C \] Since \(7A + 4B = C\) and \(7D + 4E = F\), this equality holds true. **Conclusion:** \(\boxed{\text{This system is guaranteed to have }(7,4)}\) **Final Answer:** All systems except the second one are guaranteed to have the solution \((7, 4)\).