Solve the Fraction Equation: -1/2 + (1/3)x = 1/5 + x

Q: How do I combine $ \frac{1}{3}x + x $ when one term doesn't show a fraction?

Remember that x is the same as $ \frac{1}{1}x $! To add them, convert to the same denominator: $ \frac{1}{3}x + \frac{3}{3}x = \frac{4}{3}x $.

Q: Why do I need to find a common denominator for $ \frac{1}{5} + \frac{1}{2} $ ?

You cannot add fractions until they have the same bottom number (denominator). The LCD of 5 and 2 is 10, so: $ \frac{1}{5} = \frac{2}{10} $ and $ \frac{1}{2} = \frac{5}{10} $.

Q: How do I multiply $ \frac{7}{10} \times \frac{3}{4} $ correctly?

Multiply straight across: numerator × numerator and denominator × denominator. So $ \frac{7 \times 3}{10 \times 4} = \frac{21}{40} $. No common denominator needed for multiplication!

Q: What if I get confused about which side to move terms to?

A good strategy is to move all x terms to the left and all numbers to the right. This keeps your work organized and reduces mistakes!

Q: How can I check if $ \frac{21}{40} $ is really the right answer?

Substitute back into the original equation: $ -\frac{1}{2} + \frac{1}{3} \cdot \frac{21}{40} = \frac{1}{5} - \frac{21}{40} $. Both sides should equal $ -\frac{1}{8} $ when calculated correctly.

Linear Equations with Mixed Fractional Terms

Solve for x:

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's find the value of X.

00:10 First, rearrange the equation so that X is alone on one side.

00:35 Next, combine the like terms together.

00:40 Then, find a common denominator and multiply as needed.

00:55 Now, convert any mixed numbers into fractions.

01:01 To isolate X, multiply by the reciprocal of the fraction.

01:09 Remember, multiply the numerators together and the denominators together.

01:15 And that's how we solve the problem! Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve for x:

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

Step-by-step solution

We will move the elements with the X to the left side and the elements without the X to the right side, changing the plus and minus signs accordingly.

First, we move the minus X to the left section:

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

Now we move the minus 1/2 to the right section:

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

We will find a common denominator for the fractions on the right side and reduce accordingly. Convert the mixed fraction on the left side into a simple fraction:

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

$\frac{4}{3}x=\frac{7}{10}$

Multiply by $\frac{3}{4}$ to reduce the left side:

$x=\frac{7}{10}\times\frac{3}{4}=\frac{7\times3}{10\times4}=\frac{21}{40}$

Final Answer

$\frac{21}{40}$

Key Points to Remember

Essential concepts to master this topic

Rule: Collect like terms by moving variables to one side first
Technique: $\frac{1}{3}x + x = \frac{4}{3}x$ when combining fractional coefficients
Check: Substitute $\frac{21}{40}$ : both sides equal $\frac{1}{8}$ ✓

Common Mistakes

Avoid these frequent errors

Adding fractions with different denominators incorrectly
Don't add $\frac{1}{5} + \frac{1}{2} = \frac{2}{7}$ ! This gives completely wrong results because you can't add numerators and denominators separately. Always find the LCD first: $\frac{1}{5} + \frac{1}{2} = \frac{2}{10} + \frac{5}{10} = \frac{7}{10}$ .

Practice Quiz

Test your knowledge with interactive questions

$$ x+x=8 $$

FAQ

Everything you need to know about this question

How do I combine $\frac{1}{3}x + x$ when one term doesn't show a fraction?

Remember that x is the same as $\frac{1}{1}x$ ! To add them, convert to the same denominator: $\frac{1}{3}x + \frac{3}{3}x = \frac{4}{3}x$ .

Why do I need to find a common denominator for $\frac{1}{5} + \frac{1}{2}$ ?

You cannot add fractions until they have the same bottom number (denominator). The LCD of 5 and 2 is 10, so: $\frac{1}{5} = \frac{2}{10}$ and $\frac{1}{2} = \frac{5}{10}$ .

How do I multiply $\frac{7}{10} \times \frac{3}{4}$ correctly?

Multiply straight across: numerator × numerator and denominator × denominator. So $\frac{7 \times 3}{10 \times 4} = \frac{21}{40}$ . No common denominator needed for multiplication!

What if I get confused about which side to move terms to?

A good strategy is to move all x terms to the left and all numbers to the right. This keeps your work organized and reduces mistakes!

How can I check if $\frac{21}{40}$ is really the right answer?

Substitute back into the original equation: $-\frac{1}{2} + \frac{1}{3} \cdot \frac{21}{40} = \frac{1}{5} - \frac{21}{40}$ . Both sides should equal $-\frac{1}{8}$ when calculated correctly.

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Solve the Fraction Equation: -1/2 + (1/3)x = 1/5 + x

Linear Equations with Mixed Fractional Terms

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

How do I combine $\frac{1}{3}x + x$ when one term doesn't show a fraction?

Why do I need to find a common denominator for $\frac{1}{5} + \frac{1}{2}$ ?

How do I multiply $\frac{7}{10} \times \frac{3}{4}$ correctly?

What if I get confused about which side to move terms to?

How can I check if $\frac{21}{40}$ is really the right answer?

🌟 Unlock Your Math Potential

More Questions

Simplifying and Combining Like Terms

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

Balance the Fraction Equation: Solve for X in -1/7 + 3/4x = 1/8x + 3/14

Solve for X: -8+x-3(x-2)=5(2+x)-4+3x Linear Equation

Solve for X: Simplify and Balance the Equation -8x + 5(3-x) = 6 - 4x + 3(x+1)

Solve the Fraction Equation: -1/2 + (1/3)x = 1/5 + x

Linear Equations with Mixed Fractional Terms

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

How do I combine 13x+x \frac{1}{3}x + x 31​x+x when one term doesn't show a fraction?

Why do I need to find a common denominator for 15+12 \frac{1}{5} + \frac{1}{2} 51​+21​?

How do I multiply 710×34 \frac{7}{10} \times \frac{3}{4} 107​×43​ correctly?

What if I get confused about which side to move terms to?

How can I check if 2140 \frac{21}{40} 4021​ is really the right answer?

🌟 Unlock Your Math Potential

More Questions

Simplifying and Combining Like Terms

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

Balance the Fraction Equation: Solve for X in -1/7 + 3/4x = 1/8x + 3/14

Solve for X: -8+x-3(x-2)=5(2+x)-4+3x Linear Equation

Solve for X: Simplify and Balance the Equation -8x + 5(3-x) = 6 - 4x + 3(x+1)

How do I combine $\frac{1}{3}x + x$ when one term doesn't show a fraction?

Why do I need to find a common denominator for $\frac{1}{5} + \frac{1}{2}$ ?

How do I multiply $\frac{7}{10} \times \frac{3}{4}$ correctly?

How can I check if $\frac{21}{40}$ is really the right answer?