We are tasked with solving the equation:

\[
\frac{12x + 3}{5x + 6} = \frac{1}{2}
\]

### Step 1: Eliminate the denominators by cross-multiplying.

Cross-multiply to get rid of the fractions:

\[
2(12x + 3) = 1(5x + 6)
\]

### Step 2: Expand both sides.

Expand both sides of the equation:

\[
24x + 6 = 5x + 6
\]

### Step 3: Move all terms involving \(x\) to one side.

Subtract \(5x\) from both sides:

\[
24x - 5x + 6 = 6
\]

This simplifies to:

\[
19x + 6 = 6
\]

### Step 4: Solve for \(x\).

Now, subtract 6 from both sides:

\[
19x = 0
\]

Finally, divide both sides by 19:

\[
x = 0
\]

### Final Answer:

\[
x = 0
\]

Would you like more details on this process or have any further questions?

Here are 5 related questions:
1. What happens if you get an equation like \( \frac{ax+b}{cx+d} = k \)?
2. How do you solve systems of equations with fractions?
3. Can you explain the method of cross-multiplication more deeply?
4. How would the solution change if the equation involved exponents?
5. How would you verify the solution by plugging \(x = 0\) back into the original equation?

**Tip:** Always check your solution by substituting it back into the original equation to ensure it satisfies the equation!

Discover how to solve the rational equation (12x+3)/(5x+6)=1/2 step by step using cross-multiplication. This problem involves key algebraic techniques like eliminating denominators and solving linear equations. Ideal for students in Grades 8-10.

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