Find the values of the missing information. x = _______ y = _______

Steps to Solve

1. Identify Angles of the Triangle

The triangle has the angles:

• First angle: $3x$
• Second angle: $ (3x)^\circ$
• Third angle: $(2y - 6)^\circ$

According to the triangle angle sum property: $$ 3x + 3x + (2y - 6) = 180 $$

2. Set Up the Equation

Combine like terms in the angle equation: $$ 6x + (2y - 6) = 180 $$

This simplifies to: $$ 6x + 2y - 6 = 180 $$

3. Rearrange the Equation

Add 6 to both sides to isolate terms involving $x$ and $y$: $$ 6x + 2y = 186 $$

4. Solve for One Variable in Terms of the Other

Divide everything by 2 to simplify: $$ 3x + y = 93 $$

Let's express $y$ in terms of $x$: $$ y = 93 - 3x $$

5. Use Triangle Side Lengths

Next, apply the Law of Sines for the side opposite the angles. Given sides are:

• Opposite the angle $(3x)^\circ$: $8$ in
• Opposite the angle $(2y - 6)^\circ$: $(2y - 6)$ in

The Law of Sines gives: $$ \frac{8}{\sin(3x)} = \frac{2y - 6}{\sin(2y - 6)} $$

6. Substitute for y

Now substitute $y$ from the previous equation: $$ y = 93 - 3x $$

So: $$ 2y - 6 = 2(93 - 3x) - 6 = 186 - 6 - 6x = 180 - 6x $$

7. Revisit the Law of Sines Equation

Now the equation becomes: $$ \frac{8}{\sin(3x)} = \frac{180 - 6x}{\sin(2(93 - 3x) - 6)} $$

8. Solve for x and y

This is generally complex and may require numerical methods or calculators to solve. But once you have $x$, substitute back to find $y$.