https://file.aiquickdraw.com/mathbot/file/1728591314124tcz1bdr0.jpg

Let's break down the equation step by step and complete the table in the image:

### The given equation:
\[
5(2x + 1) - 2x = 10x - 2\left(x - \frac{1}{4}\right)
\]

### Step-by-Step Solution:

1. **Eliminate any parentheses**:  
   Apply the distributive property to remove parentheses.

   Equation:  
   \[
   5(2x + 1) = 10x + 5 \quad \text{and} \quad -2(x - \frac{1}{4}) = -2x + \frac{1}{2}
   \]
   So, the new equation becomes:  
   \[
   10x + 5 - 2x = 10x - 2x + \frac{1}{2}
   \]  
   Property: Distributive Property.

2. **Combine any like terms**:  
   Combine terms on both sides where possible.

   Equation:  
   \[
   (10x - 2x + 5) = (10x - 2x + \frac{1}{2})
   \]
   Simplified to:  
   \[
   8x + 5 = 8x + \frac{1}{2}
   \]  
   Property: Combining Like Terms.

3. **Eliminate the variable from the right side of the equation**:  
   Subtract \(8x\) from both sides to move all terms with \(x\) to one side.

   Equation:  
   \[
   8x - 8x + 5 = 8x - 8x + \frac{1}{2}
   \]
   Simplified to:  
   \[
   5 = \frac{1}{2}
   \]  
   Property: Subtraction Property of Equality.

4. **Combine any like terms**:  
   There are no more like terms involving \(x\) since we eliminated them in the previous step. The equation now has no variables, just constants.

   Equation:  
   \[
   5 = \frac{1}{2}
   \]  
   Property: N/A (no further combination needed).

5. **Isolate the variable**:  
   This is a contradiction because \(5 \neq \frac{1}{2}\), so there is no solution. Thus, we conclude:

   **No solution**.

6. **Solution Set**:  
   \(\emptyset\) (empty set), meaning there is no solution.

---

This contradiction indicates that the equation is inconsistent and has no solution.

Would you like further details on any of the steps? Here are five related questions to expand on this problem:

1. What does it mean when an equation has no solution?
2. Can you provide an example of an equation with infinitely many solutions?
3. How do we handle equations involving fractions and variables on both sides?
4. What is the Subtraction Property of Equality?
5. How does the Distributive Property help in solving equations?

**Tip:** Always verify your final equation for consistency. If you reach a statement like \(5 = \frac{1}{2}\), it's a clear sign the equation has no solution!

Solve the equation: 5(2x + 1) - 2x = 10x - 2(x - 1/4)

Learn how to solve the linear equation 5(2x + 1) - 2x = 10x - 2(x - 1/4) step by step. This problem involves key algebraic concepts such as the distributive property, combining like terms, and the subtraction property of equality. Suitable for students in Grade 9-10.

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