(9) Намерете неизвестния член на пропорцията. a) \( \frac{7}{15}=\frac{x}{3} \) б) \( 9: x=36: 52 \) в) \( 5: x=2,5: 8 \) г) \( x: 1,2=\frac{35}{4}: 2 \) (10) Ако \( \frac{5}{x}=\frac{10}{13} \) и \( \frac{x}{10}=\frac{y}{12} \), намерете \( x+y \). (11) Сграда има правоъгълна основа с дължина 50 m и широчина 20 m . Можете ли да начертаете в тетрадките си план на основата на сградата, ако използвате мащаб \( 1: 50 \) ? Изберете подходящ мащаб и начертайте правоъгълника, отговарящ на основата на сградата.

Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}\frac{5}{x}=\frac{10}{13}\\\frac{x}{10}=\frac{y}{12}\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=\frac{13}{2}\\\frac{x}{10}=\frac{y}{12}\end{array}\right.\) - step2: Substitute the value of \(x:\) \(\frac{\frac{13}{2}}{10}=\frac{y}{12}\) - step3: Simplify: \(\frac{13}{20}=\frac{y}{12}\) - step4: Swap the sides: \(\frac{y}{12}=\frac{13}{20}\) - step5: Multiply both sides of the equation by \(12:\) \(\frac{y}{12}\times 12=\frac{13}{20}\times 12\) - step6: Multiply the terms: \(y=\frac{13\times 12}{20}\) - step7: Evaluate: \(y=\frac{39}{5}\) - step8: Calculate: \(\left\{ \begin{array}{l}x=\frac{13}{2}\\y=\frac{39}{5}\end{array}\right.\) - step9: Check the solution: \(\left\{ \begin{array}{l}x=\frac{13}{2}\\y=\frac{39}{5}\end{array}\right.\) - step10: Rewrite: \(\left(x,y\right) = \left(\frac{13}{2},\frac{39}{5}\right)\) Solve the system of equations \( 7/15=x/3; 9/x=36/52; 5/x=2.5/8; x/1.2=35/4/2 \). Solve the system of equations by following steps: - step0: Solution: \(\left\{ \begin{array}{l}\frac{7}{15}=\frac{x}{3}\\\frac{9}{x}=\frac{36}{52}\\\frac{5}{x}=\frac{2.5}{8}\\\frac{x}{1.2}=\frac{\frac{35}{4}}{2}\end{array}\right.\) - step1: Calculate: \(\left\{ \begin{array}{l}x=\frac{7}{5}\\x=13\\x=16\\x=\frac{21}{4}\end{array}\right.\) - step2: Find the intersection: \(x \in \varnothing \) Let's solve the problems step by step. ### Problem 9: Find the unknown member of the proportion. a) \( \frac{7}{15}=\frac{x}{3} \) To find \( x \), we can cross-multiply: \[ 7 \cdot 3 = 15 \cdot x \implies 21 = 15x \implies x = \frac{21}{15} = \frac{7}{5} \] b) \( 9: x=36: 52 \) This can be written as: \[ \frac{9}{x} = \frac{36}{52} \] Cross-multiplying gives: \[ 9 \cdot 52 = 36 \cdot x \implies 468 = 36x \implies x = \frac{468}{36} = 13 \] c) \( 5: x=2.5: 8 \) This can be written as: \[ \frac{5}{x} = \frac{2.5}{8} \] Cross-multiplying gives: \[ 5 \cdot 8 = 2.5 \cdot x \implies 40 = 2.5x \implies x = \frac{40}{2.5} = 16 \] d) \( x: 1.2=\frac{35}{4}: 2 \) This can be written as: \[ \frac{x}{1.2} = \frac{35/4}{2} \] Cross-multiplying gives: \[ x \cdot 2 = 1.2 \cdot \frac{35}{4} \implies 2x = \frac{42}{4} \implies x = \frac{21}{4} = 5.25 \] ### Summary of Results for Problem 9: - a) \( x = \frac{7}{5} \) - b) \( x = 13 \) - c) \( x = 16 \) - d) \( x = 5.25 \) --- ### Problem 10: If \( \frac{5}{x}=\frac{10}{13} \) and \( \frac{x}{10}=\frac{y}{12} \), find \( x+y \). From the first equation: \[ \frac{5}{x} = \frac{10}{13} \implies 5 \cdot 13 = 10 \cdot x \implies 65 = 10x \implies x = \frac{65}{10} = 6.5 \] Now substituting \( x \) into the second equation: \[ \frac{6.5}{10} = \frac{y}{12} \implies 6.5 \cdot 12 = 10y \implies 78 = 10y \implies y = \frac{78}{10} = 7.8 \] Now, we find \( x + y \): \[ x + y = 6.5 + 7.8 = 14.3 \] ### Summary of Result for Problem 10: - \( x + y = 14.3 \) --- ### Problem 11: Drawing the plan of the building's base. The building has a rectangular base with dimensions: - Length = 50 m - Width = 20 m Using a scale of \( 1:50 \): - Length in the drawing = \( \frac{50}{50} = 1 \) m - Width in the drawing = \( \frac{20}{50} = 0.4 \) m You can draw a rectangle with dimensions 1 m by 0.4 m on your paper to represent the base of the building. ### Summary of Result for Problem 11: - Draw a rectangle with dimensions 1 m (length) and 0.4 m (width) using a scale of \( 1:50 \).