The image shows a worksheet titled "Algebra Proofs HW Worksheet" with two equations presented for algebraic proof:

1. $2y + 4 - 5y = y - 1$
2. $-3 \left( \frac{1}{2}(x - 2) \right) + 4 = -18$

Let's solve these one by one.

1. $2y + 4 - 5y = y - 1$

Step-by-step solution:

1. Combine like terms on the left side: $2y - 5y = -3y$. $-3y + 4 = y - 1$
2. Move all terms involving $y$ to one side: subtract $y$ from both sides. $-3y - y + 4 = -1$ $-4y + 4 = -1$
3. Move constants to the other side by subtracting 4 from both sides. $-4y = -5$
4. Solve for $y$ by dividing both sides by -4. $y = \frac{-5}{-4} = \frac{5}{4}$

So, the solution is $y = \frac{5}{4}$.

2. $-3 \left( \frac{1}{2}(x - 2) \right) + 4 = -18$

Step-by-step solution:

1. Distribute the -3 across the parentheses: $-3 \times \frac{1}{2}(x - 2) = \frac{-3}{2}(x - 2)$ This simplifies to: $\frac{-3}{2}x + 3 = -18$ Now we have: $\frac{-3}{2}x + 3 + 4 = -18$
2. Combine constants on the left-hand side: $\frac{-3}{2}x + 7 = -18$
3. Move the constant to the other side by subtracting 7: $\frac{-3}{2}x = -25$
4. Solve for $x$ by multiplying both sides by $\frac{-2}{3}$: $x = -25 \times \frac{-2}{3} = \frac{50}{3}$

So, the solution is $x = \frac{50}{3}$.

Would you like more details or have any questions?

Here are some related questions to explore further:

1. How can you check if your solution to an equation is correct?
2. What are like terms, and why is it important to combine them first?
3. How do you deal with fractional coefficients in algebraic equations?
4. What is the purpose of algebraic proofs in geometry?
5. How do you interpret and simplify expressions with nested parentheses?

Tip: Always double-check your work by substituting your solution back into the original equation to verify!