answer:Given the system of equations in terms of x and y: [ left{ begin{array}{l} x + 2y = k - 1 quad (1) 2x + y = 5k + 4 quad (2) end{array} right. ] To find the value of k given that the solution satisfies x + y = 5, we proceed as follows: First, we manipulate equations (1) and (2) to eliminate one of the variables. We can do this by multiplying equation (2) by 2 and then subtracting equation (1) from the result: [ 2 times (2x + y = 5k + 4) - (x + 2y = k - 1) ] This simplifies to: [ 4x + 2y - x - 2y = 10k + 8 - k + 1 ] Further simplification gives us: [ 3x = 9k + 9 ] Solving for x yields: [ x = frac{9k + 9}{3} = 3k + 3 ] Next, we substitute x = 3k + 3 into equation (1): [ 3k + 3 + 2y = k - 1 ] Solving for y gives: [ 2y = k - 1 - 3k - 3 = -2k - 4 ] [ y = frac{-2k - 4}{2} = -k - 2 ] Given that x + y = 5, we substitute x = 3k + 3 and y = -k - 2: [ 3k + 3 - k - 2 = 5 ] Simplifying this equation: [ 2k + 1 = 5 ] [ 2k = 4 ] [ k = frac{4}{2} = 2 ] Therefore, the value of k that satisfies the given conditions is boxed{2}.

answer:Amy cut off 11 inches - 7 inches = boxed{4} inches of hair.

answer:1. The function f(x) = 4sin(x)cos(x - π/6) = 4sin(x)(√3/2 cos(x) + 1/2 sin(x)) = 4(√3/4 sin(2x) + 1/2 * (1 - cos(2x))/2) = 2sin(2x - π/6) + 1. Let 2kπ - π/2 ≤ 2x - π/6 ≤ 2kπ + π/2, we get kπ - π/6 ≤ x ≤ kπ + π/3. Thus, the increasing interval of the function is [kπ - π/6, kπ + π/3], where k ∈ Z. 2. In triangle ABC, if f(A/2) = 2sin(A - π/6) + 1 = 1, then sin(A - π/6) = 0. Therefore, A = π/6. Since a = 2, the area of triangle ABC is 1/2 * b * c * sin(A) = bc/4. According to the cosine theorem, a² = 4 = b² + c² - 2bc * cos(A) = b² + c² - √3bc ≥ 2bc - √3bc. Hence, bc ≤ 4 / (2 - √3) = 4(2 + √3). Therefore, the area of triangle ABC is bc/4 ≤ 2 + √3. Thus, the maximum area of triangle ABC is boxed{2 + sqrt{3}}.

answer:1. **Define the problem and setup:** Let ( ABCDEF ) be a regular hexagon with side length ( s ). Let ( G, H, I, J, K, ) and ( L ) be the midpoints of sides ( AB, BC, CD, DE, EF, ) and ( FA ), respectively. The segments ( AH, BI, CJ, DK, EL, ) and ( FG ) bound a smaller regular hexagon inside ( ABCDEF ). 2. **Determine the coordinates of the vertices:** Assume the hexagon is centered at the origin ( O ) of the coordinate plane. The vertices of the hexagon can be represented as: [ A = (s, 0), quad B = left(frac{s}{2}, frac{ssqrt{3}}{2}right), quad C = left(-frac{s}{2}, frac{ssqrt{3}}{2}right), quad D = (-s, 0), quad E = left(-frac{s}{2}, -frac{ssqrt{3}}{2}right), quad F = left(frac{s}{2}, -frac{ssqrt{3}}{2}right) ] 3. **Find the midpoints of the sides:** The midpoints ( G, H, I, J, K, ) and ( L ) are: [ G = left(frac{3s}{4}, frac{ssqrt{3}}{4}right), quad H = left(0, frac{ssqrt{3}}{2}right), quad I = left(-frac{3s}{4}, frac{ssqrt{3}}{4}right), quad J = left(-frac{3s}{4}, -frac{ssqrt{3}}{4}right), quad K = left(0, -frac{ssqrt{3}}{2}right), quad L = left(frac{3s}{4}, -frac{ssqrt{3}}{4}right) ] 4. **Calculate the side length of the smaller hexagon:** The smaller hexagon is formed by connecting the midpoints of the sides of the larger hexagon. The distance between two adjacent midpoints (e.g., ( G ) and ( H )) is: [ GH = sqrt{left(frac{3s}{4} - 0right)^2 + left(frac{ssqrt{3}}{4} - frac{ssqrt{3}}{2}right)^2} = sqrt{left(frac{3s}{4}right)^2 + left(-frac{ssqrt{3}}{4}right)^2} = sqrt{frac{9s^2}{16} + frac{3s^2}{16}} = sqrt{frac{12s^2}{16}} = frac{ssqrt{3}}{2} ] 5. **Determine the ratio of the areas:** The area of a regular hexagon with side length ( s ) is given by: [ text{Area} = frac{3sqrt{3}}{2} s^2 ] For the smaller hexagon, the side length is ( frac{ssqrt{3}}{2} ), so its area is: [ text{Area}_{text{small}} = frac{3sqrt{3}}{2} left(frac{ssqrt{3}}{2}right)^2 = frac{3sqrt{3}}{2} cdot frac{3s^2}{4} = frac{9sqrt{3}}{8} s^2 ] 6. **Calculate the ratio of the areas:** [ frac{text{Area}_{text{small}}}{text{Area}_{text{large}}} = frac{frac{9sqrt{3}}{8} s^2}{frac{3sqrt{3}}{2} s^2} = frac{frac{9sqrt{3}}{8}}{frac{3sqrt{3}}{2}} = frac{9sqrt{3}}{8} cdot frac{2}{3sqrt{3}} = frac{9}{12} = frac{3}{4} ] 7. **Express the ratio as a fraction and find ( m + n ):** The ratio of the areas is ( frac{3}{4} ), where ( m = 3 ) and ( n = 4 ). Thus, ( m + n = 3 + 4 = 7 ). The final answer is ( boxed{7} )