Let's start by recalling the formulas for the sum of interior angles of a triangle and a hexagon.
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Sum of interior angles of a triangle: The formula is given by: \[ (n - 2) \times 180^\circ \] where \( n \) is the number of sides. For a triangle, \( n = 3 \): \[ \text{Sum of interior angles of a triangle} = (3 - 2) \times 180 = 1 \times 180 = 180^\circ \]
According to the problem, this is also given by \( 10x - 2y \): \[ 10x - 2y = 180 \quad \text{(1)} \]
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Sum of interior angles of a hexagon: Similarly, for a hexagon, \( n = 6 \): \[ \text{Sum of interior angles of a hexagon} = (6 - 2) \times 180 = 4 \times 180 = 720^\circ \]
This is represented as \( 30x + 24y \): \[ 30x + 24y = 720 \quad \text{(2)} \]
Now, we have the following system of equations to solve: \[ 10x - 2y = 180 \quad \text{(1)} \] \[ 30x + 24y = 720 \quad \text{(2)} \]
We can simplify equation (1) first. Divide all terms by 2: \[ 5x - y = 90 \quad \text{(3)} \]
Next, we can express \( y \) in terms of \( x \) from equation (3): \[ y = 5x - 90 \quad \text{(4)} \]
Now substitute equation (4) into equation (2): \[ 30x + 24(5x - 90) = 720 \] Expanding this gives: \[ 30x + 120x - 2160 = 720 \] Combining like terms: \[ 150x - 2160 = 720 \] Adding 2160 to both sides: \[ 150x = 2880 \] Now divide by 150: \[ x = \frac{2880}{150} = 19.2 \]
Now we can substitute \( x \) back into equation (4) to solve for \( y \): \[ y = 5(19.2) - 90 \] Calculating this gives: \[ y = 96 - 90 = 6 \]
Thus, the values of \( x \) and \( y \) are: \[ x = 19.2, \quad y = 6 \]
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