How to Solve Algebraic Equations with Fractions: Step-by-Step Guide
Dealing with algebraic equations that have fractions might seem daunting, but with a systematic approach, you can solve them effectively. Let's explore a detailed solution to the equation (frac{x}{4} 3 frac{1}{2} left(1 - frac{x}{3}right)). This guide will break down the process into easy-to-follow steps, making it accessible even if you're new to algebra.
Step 1: Distribute the Right Side of the Equation
The first step is to distribute any terms on the right side of the equation. In this case, we have a fraction multiplied by a difference. The equation becomes:
[frac{x}{4} 3 frac{1}{2} - frac{x}{6}]Step 2: Eliminate the Fractions
Next, to eliminate the fractions, we need to find a common denominator. In this case, the common denominator is 12. We multiply both sides of the equation by 12 to clear the fractions:
[12 left(frac{x}{4}right) - 12 cdot 3 12 left(frac{1}{2}right) - 12 left(frac{x}{6}right)]Simplifying this, we obtain:
[3x - 36 6 - 2x]Step 3: Combine Like Terms
Now, let's combine like terms by adding (2x) to both sides of the equation:
[3x - 36 2x 6 - 2x 2x]This simplifies to:
[5x - 36 6]Step 4: Isolate the Variable
To isolate (x), we add 36 to both sides of the equation:
[5x - 36 36 6 36]This simplifies to:
[5x 42]Finally, divide both sides by 5:
[x frac{42}{5} -6]Thus, the solution to the equation is:
[x -6]Alternative Method
If you encounter the equation (frac{x}{4} - 3 frac{1}{2}left(1 - frac{x}{3}right)), you can follow these steps:
Step 1: Distribute the Right Side
Distribute and simplify the right side:
[frac{x}{2} - 6 1 - frac{x}{3}]Step 2: Eliminate the Denominators
Multiply both sides by 6 to eliminate the denominators:
[6 left(frac{x}{2}right) - 6 cdot 6 6 cdot 1 - 6 left(frac{x}{3}right)]This simplifies to:
[3x - 36 6 - 2x]Step 3: Combine Like Terms
Add (2x) to both sides of the equation:
[3x - 36 2x 6 - 2x 2x]This simplifies to:
[5x - 36 6]Step 4: Isolate the Variable
Add 36 to both sides:
[5x - 36 36 6 36]This simplifies to:
[5x 42]Divide both sides by 5:
[x frac{42}{5} -6]Thus, the solution is:
[x -6]Conclusion
In conclusion, solving algebraic equations with fractions is a matter of following structured steps. Whether you use the method of distributing and eliminating fractions or the conventional approach, the key is to remain patient and organized. Practice solving similar equations to gain confidence in handling more complex algebraic expressions.