Solve the Fraction Equation with Negative Fractions: Finding X in -1/5(x-1/3)+1/15=-3/5x+1/10

Q: Why do we need to find the LCD of all denominators?

The LCD eliminates all fractions at once, making the equation much easier to solve! For denominators 5, 15, and 10, the LCD is 30 because it's the smallest number divisible by all three.

Q: How do I distribute a negative fraction correctly?

Multiply the negative fraction by each term inside the parentheses: $ -\frac{1}{5}(x - \frac{1}{3}) = -\frac{1}{5} \cdot x + (-\frac{1}{5}) \cdot (-\frac{1}{3}) $. Remember: negative times negative equals positive!

Q: What if I get a negative answer like this problem?

Negative solutions are completely normal! The value $ x = -\frac{1}{12} $ is correct. Always check your work by substituting back into the original equation.

Q: Can I solve this without clearing fractions first?

Yes, but it's much harder! Working with fractional coefficients throughout makes calculations complex and error-prone. The LCD method simplifies everything into whole numbers first.

Q: How do I combine fractions on the same side?

Find a common denominator first. In this problem, $ \frac{1}{15} + \frac{1}{15} = \frac{2}{15} $ because they already have the same denominator. Then proceed with the LCD method.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:11 Let's find the value of X together.

00:15 First, open the brackets and multiply each term properly.

00:31 Now, let's collect like terms to simplify.

00:41 Next, rearrange the equation to get X by itself on one side.

01:12 Gather similar terms again to keep it neat.

01:20 Find a common denominator, and multiply each term accordingly.

01:40 Isolate X by doing the right operations.

01:49 And that's how we solve this problem! Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Solve for X:

$-\frac{1}{5}(x-\frac{1}{3})+\frac{1}{15}=-\frac{3}{5}x+\frac{1}{10}$

2

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Distribute $-\frac{1}{5}$ across $(x-\frac{1}{3})$ .
Step 2: Clear fractions by multiplying through by the least common multiple (LCM) of the denominators.
Step 3: Simplify and combine like terms to isolate $x$ .

Let's work through each step:

Step 1: Distribute $-\frac{1}{5}$ across $(x-\frac{1}{3})$ .
$-\frac{1}{5}(x-\frac{1}{3}) = -\frac{1}{5}x + \frac{1}{15}$ .

The equation now is:
$-\frac{1}{5}x + \frac{1}{15} + \frac{1}{15} = -\frac{3}{5}x + \frac{1}{10}$ .

Simplify the left side:
$-\frac{1}{5}x + \frac{2}{15} = -\frac{3}{5}x + \frac{1}{10}$ .

Step 2: Multiply through by 30, which is the LCM of 5, 15, and 10, to clear fractions.
$30(-\frac{1}{5}x) + 30(\frac{2}{15}) = 30(-\frac{3}{5}x) + 30(\frac{1}{10})$ .

This gives us:
$-6x + 4 = -18x + 3$ .

Step 3: Solve for $x$ .
Add $18x$ to both sides to get:
$12x + 4 = 3$ .

Subtract 4 from both sides:
$12x = -1$ .

Divide both sides by 12:
$x = -\frac{1}{12}$ .

Therefore, the solution to the problem is $x = -\frac{1}{12}$ .

3

Final Answer

$-\frac{1}{12}$

Key Points to Remember

Essential concepts to master this topic

Distribution Rule: Apply negative fraction to each term inside parentheses
LCD Method: Multiply by 30 to clear denominators 5, 15, 10
Verification: Substitute $x = -\frac{1}{12}$ back into original equation ✓

Common Mistakes

Avoid these frequent errors

Incorrectly distributing the negative fraction
Don't multiply $-\frac{1}{5}$ by only the first term in parentheses = missing the second term! This gives $-\frac{1}{5}x$ instead of $-\frac{1}{5}x + \frac{1}{15}$ . Always distribute to every term inside the parentheses, keeping track of positive and negative signs.

Practice Quiz

Test your knowledge with interactive questions

$$ -16+a=-17 $$

FAQ

Everything you need to know about this question

Why do we need to find the LCD of all denominators?

+

The LCD eliminates all fractions at once, making the equation much easier to solve! For denominators 5, 15, and 10, the LCD is 30 because it's the smallest number divisible by all three.

How do I distribute a negative fraction correctly?

+

Multiply the negative fraction by each term inside the parentheses: $-\frac{1}{5}(x - \frac{1}{3}) = -\frac{1}{5} \cdot x + (-\frac{1}{5}) \cdot (-\frac{1}{3})$ . Remember: negative times negative equals positive!

What if I get a negative answer like this problem?

+

Negative solutions are completely normal! The value $x = -\frac{1}{12}$ is correct. Always check your work by substituting back into the original equation.

Can I solve this without clearing fractions first?

+

Yes, but it's much harder! Working with fractional coefficients throughout makes calculations complex and error-prone. The LCD method simplifies everything into whole numbers first.

How do I combine fractions on the same side?

+

Find a common denominator first. In this problem, $\frac{1}{15} + \frac{1}{15} = \frac{2}{15}$ because they already have the same denominator. Then proceed with the LCD method.

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Solve the Fraction Equation with Negative Fractions: Finding X in -1/5(x-1/3)+1/15=-3/5x+1/10

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