To solve the equation:

$\frac{1}{4}x + \frac{5}{3} = x + 3$

Let's first simplify and solve for $x$:

Step 1: Eliminate the fractions

To eliminate the fractions, we need to find a common denominator. The denominators are 4, 3, and 1. The least common denominator (LCD) of 4 and 3 is 12. Multiply both sides of the equation by 12 to clear the fractions:

$12 \times \left( \frac{1}{4}x + \frac{5}{3} \right) = 12 \times \left( x + 3 \right)$

Step 2: Distribute the 12

Now, distribute the 12 across each term:

$12 \times \frac{1}{4}x + 12 \times \frac{5}{3} = 12 \times x + 12 \times 3$

This simplifies to:

$3x + 20 = 12x + 36$

Step 3: Move like terms together

Now, let's get all terms with $x$ on one side and constants on the other side. Subtract $3x$ from both sides:

$20 = 9x + 36$

Next, subtract 36 from both sides:

$-16 = 9x$

Step 4: Solve for $x$

Now, divide both sides by 9:

$x = \frac{-16}{9}$

Final Answer:

$x = -\frac{16}{9}$

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Would you like any further clarification on this solution, or is there something else you'd like to explore?

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Related Questions:

1. How do you handle equations with different denominators?
2. What are the steps to solve a linear equation with fractions?
3. How do you solve equations where the variable appears on both sides?
4. How do you check if your solution is correct in equations like this?
5. What happens if you multiply both sides by a number that is not the least common denominator?

Tip: Always look for ways to eliminate fractions early in solving equations, as it simplifies the process significantly!