Solving Multi-Step Linear Equations with Fractions

A multi-step linear equation is an algebraic equation that contains a variable and requires more than one step to solve.

• It involves performing a sequence of operations such as addition, subtraction, multiplication, and division to isolate the variable on one side of the equation.
• These equations may also include brackets, fractions, or like terms that need to be simplified before solving.
• The main goal is to rearrange the equation step by step until the value of the variable is found.

Steps to solve

Step 1: Simplify the Fractions: Ensure all fractions are in their simplest form. For example, reduce to

Step 2: Determine the Least Common Denominator (LCD): Identify the LCD of all the fractions in the equation. The LCD is the smallest number that all denominators can divide into evenly.

Step 3: Clear the Fractions: Multiply every term in the equation by the LCD to eliminate the fractions. For example, in the equation, the LCD is 12.

Multiply every term by 12:

Step 4: Solve the Simplified Equation: Solve the resulting linear equation using standard algebraic methods. For example:

Example- Consider the equation .

Step 1: Find the least common denominator (LCD) of fractions, i.e. 30.

Step 2: Multiply each term by 30 to clear the fractions:

30(3x/5) - 30(2/3) = 30(1/2)

Step 3: Simplify: 18x - 20 = 15

Step 4: Isolate the variable x by adding 20 to both sides: 18x = 35

Step 5: Finally, divide by 18 to solve for x: x = 35/18

Solved Examples

Example 1: Solve

Find the Least Common Denominator (LCD) of 5, 3, and 2, which is equal to 30.

Multiply through by 30:

Example 2: Solve

Find the LCD of 7, 6, and 2, which is equal to 42.

Multiply through by 42:

Example 3: Solve

Find Least Common Denominator (LCD): The LCD of 3, 4, and 6 is 12

Multiply every term by 12:

Simplifies to:

8x + 3 = 10

Subtract 3 from both sides:

8x = 7

x = 7/8

Example 4: Solve

LCD of 7, 5, and 2 is 70.

Multiply every term by 70

Simplifies to:

40x - 42 = 35

Add 42 to both sides:

40x = 77

x = 77/40

Example 5: Solve

Cross-Multiply:

Simplifies to:

6x - 9 = 20

Add 9 to both sides

6x = 29

x = 29/6

Example 6: Solve

Cross-Multiply:

Simplifies to:

28x - 4 = 6x + 15

Combine Like Terms: Subtract 6x from both sides:

22x - 4 = 15

Add 4 to both sides:

22x = 19

x = 19/22

Example 7: Solve

Cross-Multiply:

Simplifies to:

18x + 12 = 20

Subtract 12 from both sides:

18x = 8

x = 8/18 = 4/9

Example 8: Solve

Cross-Multiply:

Simplifies to:

24x+4 = 6x-8

Combine Like Terms: Subtract 6x from both side:

18x+4 = -8

Subtract 4 from both sides:

18x = -12

x = -12/18 = -2/3

Example 9: Solve

Cross-Multiply:

Simplifies to:

9x-6 = 35

Add 6 to both sides:

9x = 41

x = 41/9

Example 10: Solve

Cross-Multiply:

Simplifies to:

12x + 9 = 10x -2

Combine Like Terms: Subtract 10x from both side :

2x + 9 = -11

Subtract 9 from both sides:

2x = -11

x = -11/2

Practice Problems

Problem 1: Solve:

Problem 2: Solve:

Problem 3: Solve:

Problem 4: Solve:

Problem 5: Solve:

Problem 6: Solve:

Problem 7: Solve:

Problem 8: Solve:

Problem 9: Solve:

Problem 10: Solve:

Related Articles

• Simplifying Fractions
• Least Common Denominator (LCD)
• Solving Linear Equations
• Linear Equations in One Variable