Solution: From $ a - b = 1 $, express $ a = b + 1 $. Substitute into the first equation: $ 2(b + 1) + 3b = 14 \Rightarrow 2b + 2 + 3b = 14 \Rightarrow 5b = 12 \Rightarrow b = \frac{12}{5} $. Then $ a = \frac{12}{5} + 1 = \frac{17}{5} $. Compute $ 5a + 2b = 5\left(\frac{17}{5}\right) + 2\left(\frac{12}{5}\right) = 17 + \frac{24}{5} = \frac{109}{5} $. The value is $ \boxed{\dfrac{109}{5}} $.Question: A hydrologist is modeling the cross-section of a riverbed, approximated as an isosceles trapezoid

["Solution and Calculation: Finding the Height of an Isosceles Trapezoidal Riverbed", "When modeling natural formations such as riverbeds, precise geometric descriptions are essential. A common approximation of a riverbed’s cross-section is an isosceles trapezoid, defined by two parallel bases and two equal-length legs. In this case, the trapezoid has bases of 120 meters and 180 meters, and each leg measures 50 meters. To determine the height of the trapezoid, we use algebraic geometry based on the symmetry of the shape.", "### Step-by-step Derivation", "Given:\n- Longer base: 180 m\n- Shorter base: 120 m\n- Leg length: 50 m each", "Because the trapezoid is isosceles, the non-parallel bases are centered, forming two congruent right triangles on either side when the height is dropped from the top base to the bottom base.", "1. Calculate the horizontal extension on each side\n The difference in the base lengths is $ 180 - 120 = 60 $ meters.\n Since the trapezoid is symmetric, this excess length is evenly split between the two ends:\n $$\n \ ext{Horizontal leg of each triangle} = \frac{60}{2} = 30 \ ext{ m}\n $$", "2. Apply the Pythagorean Theorem\n Each right triangle formed has:\n - Vertical leg = height $ h $ (unknown, to be found)\n - Horizontal leg = 30 m\n - Hypotenuse = leg of trapezoid = 50 m", "Using the Pythagorean theorem:\n $$\n h^2 + 30^2 = 50^2\n $$\n $$\n h^2 + 900 = 2500\n $$\n $$\n h^2 = 2500 - 900 = 1600\n $$\n $$\n h = \sqrt{1600} = 40 \ ext{ m}\n $$", "Thus, the height of the riverbed’s trapezoidal cross-section is $ 40 $ meters.", "### Verification via Algebra (Similar to Given Equation Format)", "To verify using symbolic substitution—mirroring the method seen in linear equations—let $ a = h $, $ b = 30 $, and $ c = 50 $. The relationship $ a = b + 1 $ doesn’t apply directly, but we can reframe:\nFrom geometry, we derive:\n$$\nh = \sqrt{50^2 - 30^2} = \sqrt{2500 - 900} = \sqrt{1600}\n$$\nThis confirms $ h = 40 $.", "Alternatively, inspired by equation-solving style:\nIf $ h = 30 + d $, but no linear equation needed—pure geometric reasoning suffices here due to symmetry.", "### Final Result", "The height of the trapezoidal riverbed cross-section is $ \boxed{40} \ ext{ meters} $.", "This precise computation aids hydrologists in analyzing flow dynamics, sediment deposition, and ecological habitat depth within river channels—proving how mathematical modeling supports environmental science."]