Algebraic and Geometric Problem Solutions

Problem 7 Solutions and Algebraic Steps:
- The initial equation setup evaluates to $11 = \frac{1}{2}(5x - 8)$ .
- Multiplying both sides by $2$ yields: $22 = 5x - 8$ .
- Adding $8$ to both sides isolates the linear term containing the variable $x$ : $30 = 5x$ .
- Dividing both sides by $5$ gives the final solution: $x = 6$ .
Problem 8 Solutions and Radical Steps:
- The initial algebraic radical statement is $8 = \sqrt{9x + 10}$ .
- The intermediate simplified relation provides: $16 = 9x + 10$ .
- Subtracting $10$ from both sides isolates the variable term: $6 = 9x$ .
- Reducing the fraction gives: $\frac{2}{3}$ .
- The evaluated variable solution is $x = \frac{2}{3}$ .

Problem 9 Angle Determination:
- Angle $b$ is vertical to an angle measuring $66^\circ$ , which establishes that $b = 66^\circ$ because vertical angles are equal.
- Angle $c$ forms a straight line with $66^\circ$ , indicating that they are supplementary angles that add up to $180^\circ$
- Solving for $c$ : $c = 180^\circ - 66^\circ = 114^\circ$
- Final angle results: $b = 66^\circ$ and $c = 114^\circ$
Problem 10 Angle Determination:
- Angle $c$ is vertical to $100^\circ$ , which establishes that $c = 100^\circ$
- Angle $b$ is supplementary to $100^\circ$ , which means $b = 180^\circ - 100^\circ = 80^\circ$
- Final angle results: $b = 80^\circ$ and $c = 100^\circ$

Problem 11 Angle Calculation:
- The whole corner forms a right angle measuring $90^\circ$
- The equation relating variable $a$ and the given angle is $a + 31^\circ = 90^\circ$
- Solving for $a$ gives $a = 59^\circ$
- Associated numerical value: $7$