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 \frac{4}{8} to \frac{1}{2}

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\frac{2x}{3} + \frac{1}{4} = \frac{5}{6} , the LCD is 12.

Multiply every term by 12: 12(\frac{2x}{3}) + 12(\frac{1}{4}) = 12(\frac{5}{6}) \\8x+3=10

Step 4: Solve the Simplified Equation: Solve the resulting linear equation using standard algebraic methods. For example:
8x + 3 = 10 \\8x=10-3\\x=\frac{7}{8}

Example- Consider the equation \frac{3x}{5} - \frac{2}{3} = \frac{1}{2}.

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

Example 1: Solve \frac{3x}{5} - \frac{2}{3} = \frac{1}{2}

Find the Least Common Denominator (LCD) of 5, 3, and 2, which is equal to 30.
Multiply through by 30:
30(\frac{3x}{5}) - 30(\frac{2}{3}) = 30(\frac{1}{2})
18x − 20 = 15
18x=35
x = \frac{35}{18}

Example 2: Solve \frac{4x}{7} + \frac{5}{6} = \frac{3}{2}

Find the LCD of 7, 6, and 2, which is equal to 42.
Multiply through by 42:
42(\frac{4x}{7}) + 42(\frac{5}{6}) = 42(\frac{3}{2})
24x+35=63
24x=28
x = \frac{28}{24} = \frac{7}{6}

Example 3: Solve \frac{2x}{3} + \frac{1}{4} = \frac{5}{6}

Find Least Common Denominator (LCD): The LCD of 3, 4, and 6 is 12
Multiply every term by 12:
12 \cdot \frac{2x}{3} + 12 \cdot \frac{1}{4} = 12 \cdot \frac{5}{6}
Simplifies to:
8x + 3 = 10
Subtract 3 from both sides:
8x = 7
x = 7/8

Example 4: Solve \frac{4x}{7} - \frac{3}{5} = \frac{1}{2}

LCD of 7, 5, and 2 is 70.
Multiply every term by 70
70 \cdot \frac{4x}{7} - 70 \cdot \frac{3}{5} = 70 \cdot \frac{1}{2}
Simplifies to:
40x - 42 = 35
Add 42 to both sides:
40x = 77
x = 77/40

Example 5: Solve \frac{2x-3}{5} = \frac{4}{3}

Cross-Multiply:
3(2x-3) = 4 \cdot 5
Simplifies to:
6x - 9 = 20
Add 9 to both sides
6x = 29
x = 29/6

Example 6: Solve \frac{7x-1}{3} = \frac{2x+5}{4}

Cross-Multiply:
4(7x - 1) = 3(2x+5)
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 \frac{3x+2}{4} = \frac{5}{6}

Cross-Multiply:
6(3x + 2) = 5 \cdot 4
Simplifies to:
18x + 12 = 20
Subtract 12 from both sides:
18x = 8
x = 8/18 = 4/9

Example 8: Solve \frac{6x+1}{2} = \frac{3x-4}{4}

Cross-Multiply:
4(6x+1) = 2(3x-4)
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 \frac{3x-2}{5} = \frac{7}{3}

Cross-Multiply:
3(3x-2) = 7\cdot5
Simplifies to:
9x-6 = 35
Add 6 to both sides:
9x = 41
x = 41/9

Example 10: Solve \frac{4x+3}{2} = \frac{5x-1}{3}

Cross-Multiply:
3(4x + 3) = 2(5x - 1)
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: \frac{5x-3}{7} = \frac{2x+4}{3}

Problem 2: Solve: \frac{2x+1}{5} = \frac{x-3}{2}

Problem 3: Solve: \frac{3x-2}{4} = \frac{x-3}{2}

Problem 4: Solve: \frac{4x+3}{8} = \frac{5x-2}{7}

Problem 5: Solve: \frac{7x-5}{9} = \frac{2x+6}{7}

Problem 6: Solve: \frac{6x+2}{3} = \frac{4x-1}{5}

Problem 7: Solve: \frac{5x-4}{6} = \frac{3x+2}{8}

Problem 8: Solve: \frac{8x+1}{7} = \frac{2x-3}{5}

Problem 9: Solve: \frac{9x-2}{4} = \frac{3x+5}{6}

Problem 10: Solve: \frac{10x+3}{5} = \frac{4x-7}{9}

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

Solving Multi-Step Linear Equations with Fractions

Steps to solve

Solved Examples

Practice Problems

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