You've uploaded an image with four linear equations labeled (a) through (d). Let's solve each of them one by one:

a. $3x + 7 = -x - 1$

1. Add $x$ to both sides to move all $x$-terms to one side: $3x + x + 7 = -1$ $4x + 7 = -1$

2. Subtract 7 from both sides: $4x = -1 - 7$ $4x = -8$

3. Divide both sides by 4: $x = \frac{-8}{4}$ $x = -2$

b. $1 - 2x + 5 = 4x - 3$

1. Combine constants on the left-hand side: $(1 + 5) - 2x = 4x - 3$ $6 - 2x = 4x - 3$

2. Add $2x$ to both sides: $6 = 6x - 3$

3. Add 3 to both sides: $6 + 3 = 6x$ $9 = 6x$

4. Divide both sides by 6: $x = \frac{9}{6} = \frac{3}{2}$ $x = 1.5$

c. $4x - 2 + x = -2 + 2x$

1. Combine like terms on both sides: $(4x + x) - 2 = -2 + 2x$ $5x - 2 = -2 + 2x$

2. Add 2 to both sides: $5x = 2x$

3. Subtract $2x$ from both sides: $5x - 2x = 0$ $3x = 0$

4. Divide both sides by 3: $x = \frac{0}{3} = 0$

d. $3x - 4 + 1 = -2x - 5 + 5x$

1. Combine like terms on both sides: $3x - 3 = (-2x + 5x) - 5$ $3x - 3 = 3x - 5$

2. Subtract $3x$ from both sides: $-3 = -5$ This is a false statement, which means there is no solution. This equation is inconsistent.

If you have any questions or need more details about any step, feel free to ask!

Here are five related questions for you to consider:

1. How do you handle cases where there's no solution in a system of equations?
2. What does it mean when an equation simplifies to a true statement like $0 = 0$?
3. How can you verify the solution of a linear equation?
4. What is the graphical interpretation of an equation with no solution?
5. How do you approach solving linear systems with more than two variables?

Tip: Always simplify your equations step-by-step to avoid mistakes, especially when combining like terms.