4.7 Solve Equations with Fractions

Solving Equations with Fractions

Fractions in equations follow the same rules as any other equation. The goal is still to isolate the variable. The difference is you need a few extra techniques to deal with the fractions along the way.

This section covers how to verify solutions, solve equations that contain fractions, translate word problems into fraction equations, and work with algebraic expressions.

Solving Equations with Fractions

Verification of Fraction Equations

Before solving an equation, you should know how to check whether a given value actually works. The process is straightforward: plug the value in and see if both sides come out equal.

1. Substitute the variable with the given fraction.
2. Perform the arithmetic to simplify each side.
3. Compare the two sides. If they're equal, the value is a solution. If not, it isn't.

Example: Does $x = \frac{1}{2}$ satisfy $\frac{2}{3}x + \frac{1}{4} = \frac{7}{12}$?

• Substitute: $\frac{2}{3} \cdot \frac{1}{2} + \frac{1}{4}$
• Simplify: $\frac{2}{6} + \frac{1}{4} = \frac{1}{3} + \frac{1}{4}$
• Find a common denominator: $\frac{4}{12} + \frac{3}{12} = \frac{7}{12}$
• Both sides equal $\frac{7}{12}$, so yes, $x = \frac{1}{2}$ is a solution.

Now try $x = \frac{3}{4}$: $\frac{2}{3} \cdot \frac{3}{4} + \frac{1}{4} = \frac{1}{2} + \frac{1}{4} = \frac{3}{4}$. Since $\frac{3}{4} \neq \frac{7}{12}$, this value is not a solution.

Solving Equations with Fractions

The fastest way to solve an equation full of fractions is to clear the fractions first by using the least common denominator (LCD).

Example: Solve $\frac{1}{2}x + \frac{1}{3} = \frac{1}{4}$

1. Find the LCD of all denominators in the equation. The denominators are 2, 3, and 4, so the LCD is 12.
2. Multiply every term on both sides by the LCD to eliminate the fractions: $12 \cdot \frac{1}{2}x + 12 \cdot \frac{1}{3} = 12 \cdot \frac{1}{4}$ This simplifies to: $6x + 4 = 3$
3. Isolate the variable. Subtract 4 from both sides: $6x = -1$
4. Divide both sides by the coefficient (6): $x = -\frac{1}{6}$

After clearing fractions, you're just solving a regular equation. That's why finding the LCD is so helpful.

Multiplication Property for Fraction Equations

When the equation has a single fraction multiplied by the variable (like $\frac{2}{3}x = \frac{1}{4}$), there's a quicker approach: multiply both sides by the reciprocal of the fraction coefficient.

The reciprocal of a fraction $\frac{a}{b}$ is $\frac{b}{a}$. Multiplying a fraction by its reciprocal always gives 1, which isolates the variable in one step.

Example: Solve $\frac{2}{3}x = \frac{1}{4}$

1. The coefficient of $x$ is $\frac{2}{3}$. Its reciprocal is $\frac{3}{2}$.
2. Multiply both sides by $\frac{3}{2}$: $\frac{3}{2} \cdot \frac{2}{3}x = \frac{3}{2} \cdot \frac{1}{4}$
3. Simplify. On the left, $\frac{3}{2} \cdot \frac{2}{3} = 1$, so you get: $x = \frac{3}{8}$

This method works whenever the variable has a single fraction coefficient and nothing else on that side of the equation.

Word Problems to Fraction Equations

Turning a word problem into an equation takes practice, but the steps are consistent:

1. Identify the unknown and assign it a variable. For example, let $x$ represent the number of apples.

2. Translate the words into math. Look for phrases like "half of," "one-third of," or "a quarter of" and write them as fractions.

   • "Half of the apples plus 3 equals 7" becomes $\frac{1}{2}x + 3 = 7$

3. Solve the equation using the techniques above:

   • Subtract 3 from both sides: $\frac{1}{2}x = 4$
   • Multiply both sides by 2: $x = 8$

4. Check your answer in the context of the problem. Half of 8 is 4, and $4 + 3 = 7$. That matches, so 8 apples is correct.

Always re-read the original problem after solving. Make sure your answer actually makes sense (for instance, you can't have a negative number of apples).

Working with Algebraic Expressions and Rational Equations

An algebraic expression combines variables, numbers, and operations (like $2x + 3$ or $\frac{x}{2} - 1$). Expressions don't have an equals sign; equations do.

A rational equation is an equation that contains fractions with variables in the denominator, such as $\frac{5}{x} = \frac{1}{3}$. These come up more in later courses, but the core strategy is the same: find a common denominator, clear the fractions, and solve. The techniques you've learned here for fraction equations are the foundation for handling rational equations down the road.