Solving Equations with Fractions

In this lesson, we will learn how to solve multi-step equations with fractions by clearing the denominator using the LCD.

Solving equations with fractions involves the same principles as solving equations with whole numbers.

If you prefer not to work with fractions you can first eliminate the fractions by multiplying by the least common denominator. This can simplify the arithmetic and reduce the chance of errors.

Method 1: Multiplying by the Least Common Denominator (LCD)
This is the most common and often the most efficient method.

1. Find the Least Common Denominator (LCD): Identify all the denominators in the equation. The LCD is the smallest multiple that all the denominators divide into evenly.
   
2. Multiply Every Term by the LCD: Multiply both sides of the equation (and every individual term within them) by the LCD. This will cancel out all the denominators.
   
3. Solve the Resulting Equation: After clearing the fractions, you’ll have an equation with whole numbers. Solve this equation using the standard steps (combine like terms, isolate the variable).
   
4. Check Your Solution: Substitute your answer back into the original equation (with fractions) to verify that it is correct.

The following diagram gives an example of solving fractional equation. Scroll down the page for more examples and solutions of solving fractional equations.

Algebra Worksheets
Practice your skills with the following Algebra worksheets:
Printable & Online Algebra Worksheets

Example 1: Simple Fractions
Solve: \(\frac{x}{3} + \frac{1}{2} = \frac{5}{6}\)

1. LCD: The denominators are 3, 2, and 6. The LCD of 3, 2, and 6 is 6.
   
2. Multiply Every Term by the LCD:
   \(6 \cdot \frac{x}{3} + 6 \cdot \frac{1}{2} = 6 \cdot \frac{5}{6}\)
   \(2x + 3 = 5\)
   
3. Solve the Resulting Equation:
   Subtract 3 from both sides:
   \(2x = 5 - 3\)
   \(2x = 2\)
   Divide by 2:
   \(x = 1\)
   
4. Check:
   \(\frac{1}{3} + \frac{1}{2} = \frac{2}{6} + \frac{3}{6} = \frac{5}{6}\) (Correct)
   

Example 2: Variable in the Denominator
Solve: \(\frac{2}{y} - \frac{3}{4} = \frac{5}{2y}\)

1. LCD: The denominators are \(y\), 4, and \(2y\). The LCD of \(y\), 4, and \(2y\) is \(4y\).
   
2. Multiply Every Term by the LCD:
   \(4y \cdot \frac{2}{y} - 4y \cdot \frac{3}{4} = 4y \cdot \frac{5}{2y}\)
   \(8 - 3y = 10\)
   
3. Solve the Resulting Equation:
   Subtract 8 from both sides:
   \(-3y = 10 - 8\)
   \(-3y = 2\)
   Divide by -3:
   \(y = -\frac{2}{3}\)
   
4. Check:
   \(\frac{2}{-\frac{2}{3}} - \frac{3}{4} = 2 \cdot (-\frac{3}{2}) - \frac{3}{4} = -3 - \frac{3}{4} = -\frac{12}{4} - \frac{3}{4} = -\frac{15}{4}\)
   \(\frac{5}{2(-\frac{2}{3})} = \frac{5}{-\frac{4}{3}} = 5 \cdot (-\frac{3}{4}) = -\frac{15}{4}\) (Correct)
   

Method 2: Cross-Multiplication (for proportions)
This method is shortcut for equations where one fraction is equal to another fraction (a proportion).
If \(\frac{a}{b} = \frac{c}{d}\), then \(ad = bc\).

Example 3: Using Cross-Multiplication
Solve: \(\frac{x + 2}{3} = \frac{x - 1}{4}\)

1. Proportion: We have one fraction on each side.
   
2. Cross-Multiply:
   \(4(x + 2) = 3(x - 1)\)
   
3. Solve the Resulting Equation:
   \(4x + 8 = 3x - 3\)
   \(4x - 3x = -3 - 8\)
   \(x = -11\)
   
4. Check:
   \(\frac{-11 + 2}{3} = \frac{-9}{3} = -3\)
   \(\frac{-11 - 1}{4} = \frac{-12}{4} = -3\) (Correct)
   

Solving Equations by Clearing the Denominator
This video shows how to clear the denominator in equations with fractions and the solve the equations. It will also show how to solve the equations without clearing the fractions.

• Show Step-by-step Solutions

Solving an Equation with Fractions (Clear Fractions)
This video explains how to solve a equation with fractions by first clearing the fractions.

• Show Step-by-step Solutions

Solving fractional equations by clearing denominators This video shows how to solve a proportion by cross-multiplication. It will also show how to solve fractional equations by clearing denominators.

• Show Step-by-step Solutions

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