You have provided a set of simultaneous equations. Let's solve one of them as an example. If you have a specific equation in mind or a particular problem you'd like to solve, please let me know. Otherwise, I'll proceed with solving the first problem on the list.

Problem 1:

Given the simultaneous equations: $x^2 + xy = 12$ $3x + y = 10$

Step 1: Express $y$ in terms of $x$ using the second equation. $y = 10 - 3x$

Step 2: Substitute $y = 10 - 3x$ into the first equation. $x^2 + x(10 - 3x) = 12$ $x^2 + 10x - 3x^2 = 12$ $-2x^2 + 10x = 12$

Step 3: Rearrange to form a quadratic equation. $2x^2 - 10x + 12 = 0$

Step 4: Simplify by dividing by 2. $x^2 - 5x + 6 = 0$

Step 5: Factor the quadratic equation. $(x - 2)(x - 3) = 0$

Step 6: Solve for $x$. $x = 2 \quad \text{or} \quad x = 3$

Step 7: Find the corresponding $y$ values using $y = 10 - 3x$.

For $x = 2$: $y = 10 - 3(2) = 4$

For $x = 3$: $y = 10 - 3(3) = 1$

Final Solutions:

$(x, y) = (2, 4) \quad \text{or} \quad (3, 1)$

Would you like to see the solutions for the other problems or any specific one?

Here are 5 related questions for further practice:

1. How do you solve a quadratic equation by factoring?
2. What is the general method for solving systems of equations using substitution?
3. How can simultaneous equations be solved graphically?
4. What is the significance of the discriminant in solving quadratic equations?
5. How would the solution change if the equations were non-linear or involved higher powers?

Tip: Always double-check the solutions by substituting them back into the original equations to ensure they satisfy both equations.