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.

one third y minus one fourth left parenthesis y minus 2 right parenthesis

1

3

y−

1

4

(y−2)

equals

=

4

4y
















**Question content area bottom**

**Part 1**

**​a)** The LCD is 




enter your response here

.

​(Simplify your​ answer.)

**Part 2**

**​b)** What is the new​ equation?


**A.**

4

4y

minus

−

3

3​(y

minus

−

2

2​)

equals

=

48

48y


**B.**

y

minus

−​(y

minus

−

2

2​)

equals

=

48

48y


**C.**

4

4y

minus

−

3

3​(y

minus

−

6

6​)

equals

=

48

48y


**D.**

3

3y

minus

−

4

4​(y

minus

−

2

2​)

equals

=

48

48y

**Part 3**

**​c)** What is the​ solution? Select the correct choice​ and, if​ necessary, fill in the answer box to complete your choice.


**A.**

y

equals

=




enter your response here

 ​(Type an integer or a simplified​ fraction.)


**B.**

The solution is all real numbers.


**C.**

There is no solution.


Let's solve the linear equation step by step.

### Original Equation:

\[
\frac{1}{3}y - \frac{1}{4}(y - 2) = 4y
\]

### **Part 1: Finding the LCD**

a) **Determine the LCD of all denominators in the given linear equation:**

- The denominators in the equation are \(3\) and \(4\).
- The Least Common Denominator (LCD) of \(3\) and \(4\) is \(12\).

**Answer for Part 1:** The LCD is **12**.

### **Part 2: Multiply the Equation by the LCD**

b) **Write the new linear equation obtained after multiplying both sides of the original equation by the LCD:**

- Multiply each term by \(12\):

\[
12 \cdot \frac{1}{3}y - 12 \cdot \frac{1}{4}(y - 2) = 12 \cdot 4y
\]

- Simplifying each term:

\[
4y - 3(y - 2) = 48y
\]

So, the new equation is:

**Answer for Part 2:** The new equation is **D. \(4y - 3(y - 2) = 48y\)**.

### **Part 3: Solve the Equation**

c) **Solve the equation:**

Now, let's simplify and solve the equation:

1. Distribute \(-3\) inside the parentheses:

\[
4y - 3y + 6 = 48y
\]

2. Combine like terms:

\[
y + 6 = 48y
\]

3. Move \(y\) to the right side of the equation:

\[
6 = 48y - y
\]

4. Simplify the equation:

\[
6 = 47y
\]

5. Solve for \(y\):

\[
y = \frac{6}{47}
\]

So, the solution is:

**Answer for Part 3:** **A. \(y = \frac{6}{47}\)**

---

Would you like further details or explanations on any of these steps?

### Follow-up Questions:

1. What is the process to find the LCD for more complex denominators?
2. How do you verify the solution of a linear equation after solving it?
3. What are the implications if an equation has no solution or infinite solutions?
4. How can you handle equations involving fractions without using the LCD method?
5. What are common mistakes to avoid when solving linear equations?

**Tip:** Always double-check your steps by substituting your solution back into the original equation to verify its correctness.

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.
one third y minus one fourth left parenthesis y minus 2 right parenthesis equals 4y.

Learn how to solve linear equations involving fractions through a step-by-step process. This problem includes finding the Least Common Denominator (LCD), rewriting the equation, and solving for the variable. Suitable for students in Grades 8-10.

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