Let's solve the equation:

\[
\frac{9}{4}y - 12 = \frac{1}{4}y - 4
\]

### Step 1: Eliminate the fractions
To eliminate the fractions, multiply the entire equation by 4 (the denominator of the fractions):

\[
4 \times \left(\frac{9}{4}y - 12\right) = 4 \times \left(\frac{1}{4}y - 4\right)
\]

This simplifies to:

\[
9y - 48 = y - 16
\]

### Step 2: Move the variables to one side
Now, subtract \(y\) from both sides to get all the terms involving \(y\) on one side:

\[
9y - y - 48 = -16
\]

\[
8y - 48 = -16
\]

### Step 3: Isolate the variable \(y\)
Add 48 to both sides to get rid of the constant term on the left:

\[
8y = -16 + 48
\]

\[
8y = 32
\]

### Step 4: Solve for \(y\)
Finally, divide both sides by 8:

\[
y = \frac{32}{8}
\]

\[
y = 4
\]

Thus, the solution is \(y = 4\).

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How can you solve equations with variables on both sides more efficiently?
2. What are the steps to eliminate fractions in an equation?
3. How do you handle negative coefficients when solving equations?
4. Can you explain why moving all variables to one side is a good approach in algebraic equations?
5. How would the solution change if the equation had decimals instead of fractions?

**Tip:** Always check your solution by substituting it back into the original equation to verify correctness.

Learn how to solve the linear equation \dfrac{9}{4}y - 12 = \dfrac{1}{4}y - 4 by eliminating fractions and using algebraic techniques. This detailed step-by-step guide covers key concepts in algebra and is ideal for students in Grades 7-8.

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