Solve for f. 5 - 2/5 f + 3/2 f = f - 1/2 f + 4

Steps to Solve

1. Combine like terms on the left side

Start with the equation:

$$ 5 - \frac{2}{5}f + \frac{3}{2}f = f - \frac{1}{2}f + 4 $$

Combine the terms with $f$ on the left side:

$$ -\frac{2}{5}f + \frac{3}{2}f = \left(\frac{3}{2} - \frac{2}{5}\right) f $$

To combine the fractions, find a common denominator (which is 10 in this case):

$$ \frac{3}{2} = \frac{15}{10} \quad \text{and} \quad -\frac{2}{5} = -\frac{4}{10} $$

So,

$$ \frac{15}{10} - \frac{4}{10} = \frac{11}{10} $$

The left side now becomes:

$$ 5 + \frac{11}{10} f = f - \frac{1}{2} f + 4 $$

2. Combine like terms on the right side

Now, simplify the right side by combining $f$ terms:

$$ f - \frac{1}{2}f = \frac{1}{2}f $$

So the equation now reads:

$$ 5 + \frac{11}{10} f = \frac{1}{2}f + 4 $$

3. Isolate the variable f

Rearrange the equation to move all $f$ terms to one side and constant terms to the other:

$$ 5 - 4 = \frac{1}{2}f - \frac{11}{10}f $$

This simplifies to:

$$ 1 = -\left(\frac{11}{10} - \frac{5}{10}\right)f $$

Where $\frac{1}{2}f = \frac{5}{10}f$.

So now:

$$ 1 = -\frac{6}{10} f $$

4. Solve for f

Multiply both sides by $-\frac{10}{6}$ to isolate $f$:

$$ f = -10 \cdot \frac{1}{6} = -\frac{10}{6} $$

This simplifies to:

$$ f = -\frac{5}{3} $$