Solve the Fraction Equation: -1/5x + 1/4x + 1/20x = 3/10 - 2/5 + 2/10x

Q: Why is the LCD 20 when the denominators are 5, 4, 20, and 10?

The LCD is the smallest number that all denominators divide into evenly. Since 20 ÷ 5 = 4, 20 ÷ 4 = 5, 20 ÷ 20 = 1, and 20 ÷ 10 = 2, we know 20 works perfectly!

Q: How do I combine like terms with fractions?

First, make sure all terms have the same denominator. Then add or subtract the numerators: $ -\frac{4}{20}x + \frac{5}{20}x + \frac{1}{20}x = \frac{2}{20}x $

Linear Equations with Multiple Fractional Terms

Solve for x:

$-\frac{1}{5}x+\frac{1}{4}x+\frac{1}{20}x-\frac{1}{5}=\frac{3}{10}-\frac{2}{5}+\frac{2}{10}x$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:04 Find the common denominator and multiply accordingly

00:47 Group terms

00:57 Arrange the equation so that X is isolated on one side

01:09 Group terms

01:13 Isolate X

01:20 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve for x:

$-\frac{1}{5}x+\frac{1}{4}x+\frac{1}{20}x-\frac{1}{5}=\frac{3}{10}-\frac{2}{5}+\frac{2}{10}x$

Step-by-step solution

Move similar terms to one side.
Create common denominators using the least common multiple of the different fractions.
Reduction of fractions.

Final Answer

-1

Key Points to Remember

Essential concepts to master this topic

Common Denominators: Find LCD of all denominators to clear fractions efficiently
Technique: LCD of 5, 4, 20, 10 is 20; multiply entire equation by 20
Check: Substitute x = -1: verify both sides equal $-\frac{1}{10}$ ✓

Common Mistakes

Avoid these frequent errors

Not multiplying every term by the LCD
Don't multiply only the fraction terms by 20 and forget constant terms = wrong equation! This creates an imbalanced equation where one side changes more than the other. Always multiply every single term on both sides by the LCD.

Practice Quiz

Test your knowledge with interactive questions

$$ x+x=8 $$

FAQ

Everything you need to know about this question

Why is the LCD 20 when the denominators are 5, 4, 20, and 10?

The LCD is the smallest number that all denominators divide into evenly. Since 20 ÷ 5 = 4, 20 ÷ 4 = 5, 20 ÷ 20 = 1, and 20 ÷ 10 = 2, we know 20 works perfectly!

What happens after I multiply everything by 20?

All fractions disappear! You get a simple equation with integers: $-4x + 5x + x - 4 = 6 - 8 + 4x$ . Much easier to solve!

How do I combine like terms with fractions?

First, make sure all terms have the same denominator. Then add or subtract the numerators: $-\frac{4}{20}x + \frac{5}{20}x + \frac{1}{20}x = \frac{2}{20}x$

Why is the answer negative?

Negative answers are completely normal! After simplifying, we get $2x = -2$ , so $x = -1$ . Always check by substituting back into the original equation.

Can I solve this without finding the LCD?

You could, but it's much harder! You'd need to convert each fraction separately. Using the LCD method clears all fractions at once, making the problem much simpler.

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