Let's solve both questions step by step.

Question 11: Solve the system of equations using substitution or elimination:

The system of equations is:

$x + 2y = 10$ $3x - 4y = 8$

Using substitution method:

1. From the first equation, solve for $x$: $x = 10 - 2y$

2. Substitute this expression for $x$ into the second equation: $3(10 - 2y) - 4y = 8$ Simplify: $30 - 6y - 4y = 8$ Combine like terms: $30 - 10y = 8$ Solve for $y$: $-10y = 8 - 30$ $-10y = -22$ $y = \frac{-22}{-10} = \frac{11}{5}$

3. Now substitute $y = \frac{11}{5}$ back into the first equation to find $x$: $x + 2\left(\frac{11}{5}\right) = 10$ Simplify: $x + \frac{22}{5} = 10$ Multiply through by 5 to clear the fraction: $5x + 22 = 50$ $5x = 50 - 22$ $5x = 28$ $x = \frac{28}{5}$

Thus, the solution to the system is: $x = \frac{28}{5}, \quad y = \frac{11}{5}$

Question 12: Simplify the trigonometric expression

$\frac{\sin(x) + \cos(x)}{\sin(x) - \cos(x)}$

Strategy:

We can use the identities and algebraic manipulation. Let's divide both the numerator and the denominator by $\cos(x)$:

$\frac{\sin(x) + \cos(x)}{\sin(x) - \cos(x)} = \frac{\frac{\sin(x)}{\cos(x)} + 1}{\frac{\sin(x)}{\cos(x)} - 1}$

Now, use the identity $\tan(x) = \frac{\sin(x)}{\cos(x)}$:

$= \frac{\tan(x) + 1}{\tan(x) - 1}$

Thus, the simplified form of the expression is:

$\frac{\tan(x) + 1}{\tan(x) - 1}$

Do you need further details or have any questions?

Here are five related questions:

1. Can you solve the system of equations using the elimination method?
2. How does substitution work when the system involves more complex equations?
3. What trigonometric identities would you use to simplify other types of expressions?
4. Can you derive another form of the simplified trigonometric expression?
5. How can you check if the solution to a system of equations is correct?

Tip: Always substitute the values back into the original equations to verify the solution.