Solve for m: \frac{2}{3}m - 2 + \frac{1}{6}m + \frac{3}{2} = \frac{1}{2}m

Steps to Solve

1. Combine like terms on the left side

First, we need to combine all terms involving $m$ on the left side of the equation:

$$ \frac{2}{3}m + \frac{1}{6}m - \frac{1}{2}m - 2 + \frac{3}{2} = 0 $$

2. Convert to a common denominator

To combine the fractions with $m$, we need a common denominator. The least common multiple of 3, 6, and 2 is 6. Rewrite the fractions:

• $ \frac{2}{3}m = \frac{4}{6}m $
• $ \frac{1}{6}m = \frac{1}{6}m $
• $ -\frac{1}{2}m = -\frac{3}{6}m $

Now we can combine them:

$$ \left( \frac{4}{6}m + \frac{1}{6}m - \frac{3}{6}m \right) $$

3. Simplify the left side

Combine the coefficients of $m$:

$$ \frac{4 + 1 - 3}{6}m = \frac{2}{6}m $$

This simplifies to:

$$ \frac{1}{3}m $$

The equation now looks like:

$$ \frac{1}{3}m - 2 + \frac{3}{2} = 0 $$

4. Combine constant terms

Now combine the constants on the left:

$$ -2 + \frac{3}{2} = -\frac{4}{2} + \frac{3}{2} = -\frac{1}{2} $$

So the equation is:

$$ \frac{1}{3}m - \frac{1}{2} = 0 $$

5. Isolate m

Add $\frac{1}{2}$ to both sides:

$$ \frac{1}{3}m = \frac{1}{2} $$

Next, multiply both sides by the reciprocal of $\frac{1}{3}$, which is 3:

$$ m = 3 \times \frac{1}{2} $$

6. Calculate the result

Thus:

$$ m = \frac{3}{2} $$