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

Q: Why do I need to move all x terms to one side?

Moving all x terms together lets you combine them into a single coefficient. This simplifies $ \frac{2}{5}x - \frac{1}{10}x $ into just $ \frac{3}{10}x $, making it much easier to solve!

Q: How do I subtract fractions with different denominators?

Find the least common denominator (LCD). For $ \frac{2}{5} $ and $ \frac{1}{10} $, the LCD is 10. Convert: $ \frac{2}{5} = \frac{4}{10} $, then subtract: $ \frac{4}{10} - \frac{1}{10} = \frac{3}{10} $.

Q: What does it mean to multiply by the reciprocal?

When you have $ \frac{3}{10}x = -1 $, multiply both sides by $ \frac{10}{3} $ (the reciprocal of $ \frac{3}{10} $). This cancels out the fraction: $ x = -1 \times \frac{10}{3} = -\frac{10}{3} $.

Q: Can I check my answer without substituting back?

Substitution is the best method! Plug $ x = -\frac{10}{3} $ into the original equation. If both sides equal the same value, you're correct. No shortcuts beat this reliable check!

Linear Equations with Mixed Fractional Terms

Solve for X:

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Arrange the equation so that one side has only the unknown X

00:28 Multiply by denominators to find the common denominator

00:43 Collect like terms

00:55 Multiply by the reciprocal to isolate X

01:01 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:

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

Step-by-step solution

To solve this equation, we will perform the following steps:

Step 1: Move terms involving $x$ to one side of the equation. Subtract $\frac{1}{10}x$ from both sides:

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

Step 2: Simplify the equation:

First, simplify the right-hand side:

-\frac{3}{4} - \frac{1}{4} = -\frac{4}{4} = -1

Step 3: Align terms with $x$ . Find a common denominator for the fractions on the left side:

The common denominator for $\frac{2}{5}x$ and $\frac{1}{10}x$ is 10.

\frac{4}{10}x - \frac{1}{10}x = \frac{3}{10}x

Step 4: Set up the simplified equation:

\frac{3}{10}x = -1

Step 5: Solve for $x$ by multiplying both sides by the reciprocal of the fraction's coefficient:

x = -1 \times \frac{10}{3} = -\frac{10}{3}

Step 6: Final answer check verifies against the multiple-choice option.

Therefore, the solution to the equation is $x = -\frac{10}{3}$ .

Final Answer

$-\frac{10}{3}$

Key Points to Remember

Essential concepts to master this topic

Isolation: Move all x terms to one side, constants to other
Common Denominator: Convert $\frac{2}{5}x - \frac{1}{10}x = \frac{4}{10}x - \frac{1}{10}x = \frac{3}{10}x$
Verification: Substitute $x = -\frac{10}{3}$ back into original equation: both sides equal $-\frac{1}{2}$ ✓

Common Mistakes

Avoid these frequent errors

Not finding common denominator for fractional coefficients
Don't subtract $\frac{2}{5}x - \frac{1}{10}x$ directly = wrong coefficient! You can't subtract fractions with different denominators. Always convert to common denominator first: $\frac{4}{10}x - \frac{1}{10}x = \frac{3}{10}x$ .

Practice Quiz

Test your knowledge with interactive questions

$$ -16+a=-17 $$

FAQ

Everything you need to know about this question

Why do I need to move all x terms to one side?

Moving all x terms together lets you combine them into a single coefficient. This simplifies $\frac{2}{5}x - \frac{1}{10}x$ into just $\frac{3}{10}x$ , making it much easier to solve!

How do I subtract fractions with different denominators?

Find the least common denominator (LCD). For $\frac{2}{5}$ and $\frac{1}{10}$ , the LCD is 10. Convert: $\frac{2}{5} = \frac{4}{10}$ , then subtract: $\frac{4}{10} - \frac{1}{10} = \frac{3}{10}$ .

What does it mean to multiply by the reciprocal?

When you have $\frac{3}{10}x = -1$ , multiply both sides by $\frac{10}{3}$ (the reciprocal of $\frac{3}{10}$ ). This cancels out the fraction: $x = -1 \times \frac{10}{3} = -\frac{10}{3}$ .

Can I check my answer without substituting back?

Substitution is the best method! Plug $x = -\frac{10}{3}$ into the original equation. If both sides equal the same value, you're correct. No shortcuts beat this reliable check!

Why is my answer negative?

Negative answers are completely normal in algebra! The equation led us to $x = -\frac{10}{3}$ because that's the only value that makes both sides equal. Always trust your work when you follow the steps correctly.

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