Question

The sum of interior angles of a triangle is given by(10x-2y) while that of a hexagon is given by (30x+24y). calculate the values of x and y

Answer 1

Let's start by recalling the formulas for the sum of interior angles of a triangle and a hexagon.

1. **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 \( 10x - 2y \):
\[
10x - 2y = 180 \quad \text{(1)}
\]

2. **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:
\[
x = 19.2, \quad y = 6
\]