Follow the​ Step-by-Step process to solve the linear equation.
​a) Determine the LCD of all denominators in the given linear equation.
​b) Write the new linear equation obtained after multiplying both sides of the original equation by the LCD.
​c) Solve the equation.
StartFraction x plus 6 Over 3 EndFraction minus StartFraction x minus 18 Over 4 EndFraction equals 3
Question content area bottom
Part 1
​a) The LCD is 
  
12.
​(Simplify your​ answer.)
Part 2
​b) What is the new​ equation?
A.
4​(xplus6​)minus3​(xminus18​)equals36
B.
12​(xplus6​)minus12​(xminus18​)equals36
C.
4​(xplus6​)minus3​(xminus18​)equals3
D.
​(xplus6​)minus​(xminus18​)equals36
Part 3
​c) What is the​ solution? Select the correct choice​ and, if​ necessary, fill in the answer box to complete your choice.
A.
xequals
  
enter your response here ​(Type an integer or a simplified​ fraction.)
B.
The solution is all real numbers.
C.


Let's solve the linear equation step by step.

### Given Equation:
\[
\frac{x + 6}{3} - \frac{x - 18}{4} = 3
\]

### Part 1: Determine the LCD
The denominators in the equation are 3 and 4. The Least Common Denominator (LCD) of 3 and 4 is 12.

### Part 2: Write the New Equation
Now, we'll multiply every term in the equation by the LCD (12) to eliminate the fractions:

\[
12 \times \left(\frac{x + 6}{3}\right) - 12 \times \left(\frac{x - 18}{4}\right) = 12 \times 3
\]

Simplifying each term:

\[
4(x + 6) - 3(x - 18) = 36
\]

So, the correct new equation is:
\[
\boxed{4(x + 6) - 3(x - 18) = 36}
\]

This corresponds to option **A**.

### Part 3: Solve the Equation
Now, let's solve the equation \(4(x + 6) - 3(x - 18) = 36\).

1. Distribute the 4 and -3:
   \[
   4x + 24 - 3x + 54 = 36
   \]

2. Combine like terms:
   \[
   (4x - 3x) + (24 + 54) = 36
   \]
   \[
   x + 78 = 36
   \]

3. Subtract 78 from both sides to isolate \(x\):
   \[
   x = 36 - 78
   \]
   \[
   x = -42
   \]

Thus, the solution is:
\[
\boxed{x = -42}
\]

This corresponds to option **A** with \(x = -42\).

---

Do you want more details or have any questions? Here are 5 related questions that could help deepen your understanding:

1. What would the equation look like if the LCD was incorrectly calculated as 6 instead of 12?
2. How do you check if \(x = -42\) is the correct solution by substituting it back into the original equation?
3. Why is it important to find the LCD when dealing with equations involving fractions?
4. What happens if you multiply the equation by a number other than the LCD?
5. How would the process change if the original equation had more complex fractions?

**Tip:** Always simplify fractions first, if possible, to make the process of finding the LCD easier.

Follow the Step-by-Step process to solve the linear equation.
a) Determine the LCD of all denominators in the given linear equation.
b) Write the new linear equation obtained after multiplying both sides of the original equation by the LCD.
c) Solve the equation.
\(\frac{x + 6}{3} - \frac{x - 18}{4} = 3\)

Part 1
a) The LCD is 12. (Simplify your answer.)

Part 2
b) What is the new equation?
A. 4(x + 6) - 3(x - 18) = 36
B. 12(x + 6) - 12(x - 18) = 36
C. 4(x + 6) - 3(x - 18) = 3
D. (x + 6) - (x - 18) = 36

Part 3
c) What is the solution? Select the correct choice and, if necessary, fill in the answer box to complete your choice.
A. x = (enter your response here) (Type an integer or a simplified fraction.)
B. The solution is all real numbers.

Learn how to solve a linear equation with fractions using a step-by-step process. This problem focuses on finding the least common denominator (LCD) and multiplying to eliminate fractions, followed by solving the simplified equation. Ideal for students in Grades 8-10.

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