We can solve the equation \( \frac{3}{2} (4 - 8x) + 5x = 1 - 7x \) step by step.

### Step 1: Distribute the \(\frac{3}{2}\) across the terms inside the parentheses.
\[
\frac{3}{2} \times 4 - \frac{3}{2} \times 8x = 6 - 12x
\]
Now the equation becomes:
\[
6 - 12x + 5x = 1 - 7x
\]

### Step 2: Combine like terms on the left side.
\[
6 - 12x + 5x = 6 - 7x = 1 - 7x
\]
Now the equation is:
\[
6 - 7x = 1 - 7x
\]

### Step 3: Eliminate \(-7x\) from both sides.
Add \(7x\) to both sides:
\[
6 = 1
\]

### Step 4: Analyze the result.
This equation leads to a contradiction (\(6 \neq 1\)). Therefore, there is **no solution** to this equation.

Would you like more details or have any questions about this process? Here are some related questions to deepen your understanding:

1. What are other examples of equations that have no solution?
2. How do we handle equations with fractions and variables?
3. What happens when an equation results in an identity (e.g., \(0 = 0\)) instead of a contradiction?
4. How can we tell early on if an equation will have no solution?
5. How does distributing work with fractions in more complex equations?

**Tip:** When solving equations, always combine like terms carefully before simplifying. This helps avoid mistakes when moving variables across the equation.

Learn how to solve the linear equation 3/2(4 - 8x) + 5x = 1 - 7x step by step. This problem demonstrates key algebraic concepts like the distributive property, combining like terms, and identifying when an equation has no solution. Suitable for students in Grades 8-10.

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