Lucy is working with her partner to solve the equation shown below.

12x+32=38x−6
1a.  
Which of the following should Lucy choose as the LCM to generate an equivalent equation in order to solve?
A.2
B.4
C.6
D.8

1b.  
What is Lucy’s equivalent equation after distributing the LCM?
A.12x+32=3x−48

B.4x+12=3x−48

C.4x+12=3x−6

D.x+8=3x−6


2.  
What is Lucy’s solution to 12x+32=38x−6
?
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Let's solve the problem step by step.

### Problem 1a: Choosing the LCM

The equation given is:
\[ 12x + 32 = 38x - 6 \]

For this problem, it's not necessary to find an LCM (Least Common Multiple) because the equation doesn't involve fractions. However, if we consider the possibility that the problem might involve multiplying both sides by a common factor to simplify, let's consider the coefficients.

The coefficients of \( x \) are 12 and 38. The Greatest Common Divisor (GCD) for 12 and 38 is 2, so if we were looking for a common factor to simplify the equation, we might consider multiplying or dividing by 2. But the LCM is typically used for terms involving different denominators, which we don't have here.

Given the options:
- **A. 2**
- **B. 4**
- **C. 6**
- **D. 8**

The best choice (if you had to pick one based on simplification or an approach like multiplying to clear fractions) would be **A. 2**. But it seems a bit misleading because, generally, an LCM isn't necessary here.

### Problem 1b: Equivalent Equation After Distributing the LCM

Given the choices, it looks like the problem is asking what happens if we distribute a factor through the equation.

If we consider that multiplying by the LCM (which might be the misunderstanding of simplifying by dividing by 2), we would have:

- Multiply the whole equation by \(\frac{1}{2}\) (which corresponds to multiplying by the LCM of 2):

\[
\frac{1}{2} \times (12x + 32) = \frac{1}{2} \times (38x - 6)
\]

This gives:
\[
6x + 16 = 19x - 3
\]

However, none of the options directly correspond to this transformation. Let's examine the options:

- **A. 12x + 32 = 3x - 48** (This is not correct, as multiplying by the LCM wouldn't result in such an equation.)
- **B. 4x + 12 = 3x - 48** (Incorrect transformation.)
- **C. 4x + 12 = 3x - 6** (Also incorrect.)
- **D. x + 8 = 3x - 6** (This seems closest, though still not following from a multiplication by 2.)

But let's proceed to solve the equation directly.

### Problem 2: Solving the Equation
Solve \( 12x + 32 = 38x - 6 \):

Step 1: Move all \( x \)-terms to one side:
\[
12x - 38x = -6 - 32
\]

Step 2: Simplify:
\[
-26x = -38
\]

Step 3: Divide both sides by \(-26\):
\[
x = \frac{-38}{-26} = \frac{19}{13}
\]

So, the solution is \( x = \frac{19}{13} \).

Would you like more details or explanations? Here are five related questions you might find interesting:

1. How can you verify that \( x = \frac{19}{13} \) is the correct solution for the original equation?
2. What is the general process for solving linear equations like \( 12x + 32 = 38x - 6 \)?
3. How would the solution change if the equation had fractions instead of whole numbers?
4. What are some common mistakes to avoid when solving linear equations?
5. How do you check if a solution satisfies the original equation?

**Tip:** Always double-check your algebraic manipulations to ensure no sign errors are made when moving terms across the equal sign!

Lucy is working with her partner to solve the equation shown below: 12x + 32 = 38x − 6. 1a. Which of the following should Lucy choose as the LCM to generate an equivalent equation in order to solve? A.2 B.4 C.6 D.8 1b. What is Lucy’s equivalent equation after distributing the LCM? A.12x+32=3x−48 B.4x+12=3x−48 C.4x+12=3x−6 D.x+8=3x−6 2. What is Lucy’s solution to 12x+32=38x−6?

Discover how to solve the linear equation 12x + 32 = 38x − 6 with step-by-step explanations. This problem explores algebraic concepts, including choosing the LCM and distributing it, and is ideal for students in Grades 7-9.

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