Linear Equations with Fractions
Solving linear equations with fractions involves a systematic approach. Here’s a step-by-step process to guide you through:
Step 1: Clear Fractions (if necessary)
If the equation has fractions, the first step is to clear them. Multiply every term of the equation by the least common denominator (LCD) to eliminate fractions. The LCD is the least of the common multiple of the denominators.
Step 2: Simplify the Equation
After multiplying by the LCD, simplify both sides of the equation by combining like terms.
Step 3: Isolate the Variable
Move all terms containing the variable to one side of the equation and constants to the other side. Use addition, subtraction, multiplication, and division to isolate the variable.
Step 4: Check Your Solution
Substitute the solution back into the original equation to ensure it satisfies the equation. If the left side equals the right side, your solution is correct.
Now, let’s work through an example:
Example: Solve the Equation
23�−14=56�+23
32​x−41​=65​x+32​
Step 1: Clear Fractions
Multiply every term by the LCD, which is 12 in this case.
12×(23�−14)=12×(56�+23)
12×(32​x−41​)=12×(65​x+32​)
This simplifies to:
8�−3=10�+8
8x−3=10x+8
Step 2: Simplify the Equation
Combine like terms on both sides:
8�−3=10�+8
8x−3=10x+8
Step 3: Isolate the Variable
Move all terms with
�
x to one side and constants to the other. Subtract
8�
8x from both sides and add 3 to both sides:
−3−8�=8
−3−8x=8
This simplifies to:
−8�=11
−8x=11
Now, divide by -8 to solve for
�
x:
�=−118
x=−811​
Step 4: Check Your Solution
Substitute
�=−118
x=−811​ back into the original equation to confirm the solution:
23×(−118)−14=56×(−118)+23
32​×(−811​)−41​=65​×(−811​)+32​
After simplifying both sides, you should get the same value, confirming that
�=−118
x=−811​ is the solution.
