\frac{1}{5}x+\frac{1}{4}=3\left(\frac{3}{4}x+1\right)

To solve the equation

\[
\frac{1}{5}x + \frac{1}{4} = 3 \left( \frac{3}{4}x + 1 \right),
\]

we start by expanding the right-hand side and simplifying the expression.

### Step 1: Expand the Right Side

Distribute the 3 on the right side:

\[
3 \left( \frac{3}{4}x + 1 \right) = 3 \cdot \frac{3}{4}x + 3 \cdot 1 = \frac{9}{4}x + 3.
\]

The equation now becomes:

\[
\frac{1}{5}x + \frac{1}{4} = \frac{9}{4}x + 3.
\]

### Step 2: Eliminate Fractions

To eliminate the fractions, multiply every term by the least common multiple (LCM) of the denominators (5 and 4). The LCM of 5 and 4 is 20. Multiply both sides by 20:

\[
20 \left( \frac{1}{5}x \right) + 20 \left( \frac{1}{4} \right) = 20 \left( \frac{9}{4}x \right) + 20 \cdot 3.
\]

Simplify:

\[
4x + 5 = 45x + 60.
\]

### Step 3: Solve for \(x\)

Rearrange the equation to isolate \(x\) on one side:

\[
4x - 45x = 60 - 5.
\]

Simplify:

\[
-41x = 55.
\]

Now, solve for \(x\) by dividing both sides by \(-41\):

\[
x = -\frac{55}{41}.
\]

### Final Answer

\[
x = -\frac{55}{41}.
\]

Would you like more details on any step, or do you have any other questions?

### Related Questions

1. How do you solve linear equations with fractions?
2. What is the process for finding the least common multiple of denominators?
3. How can we check the solution of a linear equation?
4. How do you handle equations with variables on both sides?
5. What are some tips for solving equations more efficiently?

**Tip:** When dealing with fractions, clearing them early by multiplying by the least common multiple can make solving the equation simpler.

\frac{1}{5}x + \frac{1}{4} = 3\left(\frac{3}{4}x + 1\right)

This tutorial provides a detailed, step-by-step guide on solving the linear equation \frac{1}{5}x + \frac{1}{4} = 3\left(\frac{3}{4}x + 1\right). Learn how to apply the distributive property, eliminate fractions using the least common multiple (LCM), and solve for x. Suitable for students in Grades 9-10.

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