Let's solve the equation \( 4(3x + 5) = 5(x + 4) + 7x \).

### Step 1: Expand both sides
Distribute the constants on both sides of the equation.

On the left side:
\[
4(3x + 5) = 4 \cdot 3x + 4 \cdot 5 = 12x + 20
\]

On the right side:
\[
5(x + 4) + 7x = 5 \cdot x + 5 \cdot 4 + 7x = 5x + 20 + 7x = 12x + 20
\]

So, the equation becomes:
\[
12x + 20 = 12x + 20
\]

### Step 2: Simplify the equation
Since both sides of the equation are exactly the same, the equation is an **identity**. This means that the equation is true for all values of \( x \).

Thus, the solution is:
\[
\text{All real numbers}
\]

### Summary:
The given equation is true for any value of \( x \), so the solution is all real numbers.

Would you like more details or have any questions?

Here are 5 related questions:
1. What is an identity equation, and how does it differ from a regular equation?
2. How do you recognize when an equation has no solution?
3. Can an equation have more than one solution without being an identity?
4. What methods can you use to verify if an equation is an identity?
5. How would the equation change if the constants on either side were different?

**Tip:** When solving equations, always double-check your work by plugging your solution back into the original equation to ensure it's correct.

Discover how to solve the linear identity equation 4(3x + 5) = 5(x + 4) + 7x. This step-by-step solution explains the distributive property, simplifying expressions, and understanding identity equations. Suitable for students in Grades 8-10.

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