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

Question image

Understand the Problem

The question is asking to solve the equation for the variable f, which involves rearranging and simplifying the given equation with fractions.

Answer

$$ f = -\frac{5}{3} $$
Answer for screen readers

The solution for ( f ) is

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

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:

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

  1. 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:

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

  1. 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:

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

  1. 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:

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

The solution for ( f ) is

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

More Information

This equation illustrates the process of combining like terms and isolating the variable. Solving equations with fractions often requires finding a common denominator, which is an important algebraic skill.

Tips

  • Failing to find a common denominator when combining fractions.
  • Incorrectly moving $f$ terms to one side or miscalculating during isolation steps.
  • Forgetting to change the sign when multiplying or dividing by a negative number.

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