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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.

Solve the system of linear equations: a. 3x + 7 = -x - 1, b. 1 - 2x + 5 = 4x - 3, c. 4x - 2 + x = -2 + 2x, d. 3x - 4 + 1 = -2x - 5 + 5x

Learn how to solve a system of linear equations with detailed step-by-step explanations. This example covers key algebraic concepts like combining like terms and solving for variables. Suitable for Grades 8-10 students looking to master solving linear equations.

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