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To solve the given problem, let's break it down step by step. ### Expression Analysis The mathematical expression to solve is: \[ \left(\frac{9a^2 - 1}{16b^2}\right) \div \left(\frac{3a - 1}{4b}\right) \text{ при } a = \frac{2}{3} \text{ и } b = -\frac{1}{12} \] This is a complex fraction that involves a division of two fractions. The division of fractions can be rewritten as multiplication by the reciprocal: \[ \frac{9a^2 - 1}{16b^2} \times \frac{4b}{3a - 1} \] #### Substitute Values for \(a\) and \(b\): Given: \[ a = \frac{2}{3} \] \[ b = -\frac{1}{12} \] First, calculate each part of the expression separately. ### Step-by-Step Calculation 1. **Calculate \(9a^2 - 1\):** \[ a^2 = \left(\frac{2}{3}\right)^2 = \frac{4}{9} \] \[ 9a^2 = 9 \times \frac{4}{9} = 4 \] \[ 9a^2 - 1 = 4 - 1 = 3 \] 2. **Calculate \(16b^2\):** \[ b^2 = \left(-\frac{1}{12}\right)^2 = \frac{1}{144} \] \[ 16b^2 = 16 \times \frac{1}{144} = \frac{16}{144} = \frac{1}{9} \] 3. **Calculate \(\frac{9a^2 - 1}{16b^2}\):** \[ \frac{9a^2 - 1}{16b^2} = \frac{3}{\frac{1}{9}} = 3 \times 9 = 27 \] 4. **Calculate \(3a - 1\):** \[ 3a = 3 \times \frac{2}{3} = 2 \] \[ 3a - 1 = 2 - 1 = 1 \] 5. **Calculate \(\frac{4b}{3a - 1}\):** \[ 4b = 4 \times \left(-\frac{1}{12}\right) = -\frac{4}{12} = -\frac{1}{3} \] \[ \frac{4b}{3a - 1} = \frac{-\frac{1}{3}}{1} = -\frac{1}{3} \] 6. **Final Step \((27) \times \left(-\frac{1}{3}\right)\):** \[ 27 \times \left(-\frac{1}{3}\right) = -9 \] ### Solve the System of Equations Now solve the system of linear equations: \[ \begin{cases} 3x + 2y = 8 \\ 4x - y = 7 \end{cases} \] 1. **Express \(y\) from the second equation:** \[ 4x - y = 7 \implies y = 4x - 7 \] 2. **Substitute \(y\) into the first equation:** \[ 3x + 2(4x - 7) = 8 \] \[ 3x + 8x - 14 = 8 \] \[ 11x - 14 = 8 \] \[ 11x = 22 \] \[ x = 2 \] 3. **Find \(y\):** \[ y = 4x - 7 = 4(2) - 7 = 8 - 7 = 1 \] ### Conclusion Thus, the solution of the system of equations is: \[ x = 2, \quad y = 1 \] Therefore, \(x + y = 2 + 1 = 3\). The final answer is \(-9\) for the expression involving \(a\) and \(b\), and \(x + y = 3\) for the system of equations.