Solve for X: Finding the Value in (1/4)x - (2/3)x + (1/8)x = 5

Q: Why is the LCD 24 and not 12?

The LCD must work for all denominators: 4, 3, and 8. While 12 works for 4 and 3, it doesn't work for 8 (since 8 doesn't divide evenly into 12). 24 is the smallest number that all three divide into evenly.

Q: How do I convert the improper fraction to a mixed number?

Divide the numerator by the denominator: $ -\frac{120}{7} $ means 120 ÷ 7 = 17 remainder 1. So the mixed number is $ -17\frac{1}{7} $.

Q: Why is the answer negative?

The coefficient of x became $ \frac{-7}{24} $ (negative), and we're solving $ \frac{-7}{24}x = 5 $. When you divide a positive number (5) by a negative fraction, the result must be negative.

Q: Can I solve this without finding the LCD?

You could solve term by term, but it's much more complicated and error-prone. The LCD method is the most efficient way to handle equations with multiple fractions.

Q: How do I check my work with such a complicated fraction?

Substitute $ x = -\frac{120}{7} $ back: $ \frac{1}{4}(-\frac{120}{7}) - \frac{2}{3}(-\frac{120}{7}) + \frac{1}{8}(-\frac{120}{7}) $. This should equal 5 when simplified.

Linear Equations with Multiple Fraction Terms

Solve for X:

$\frac{1}{4}x-\frac{2}{3}x+\frac{1}{8}x=5$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:04 We want to isolate the unknown X

00:07 Find the common denominator

00:13 Isolate the unknown X

00:26 Division by a fraction is multiplication by its reciprocal

00:32 Make sure to multiply numerator by numerator and denominator by denominator

00:45 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:

$\frac{1}{4}x-\frac{2}{3}x+\frac{1}{8}x=5$

Step-by-step solution

Solve the equation by following these steps:

Step 1: Find the least common multiple (LCM) of the denominators $4$ , $3$ , and $8$ . The LCM is $24$ .
Step 2: Rewrite each term so that the denominator is $24$ .
$\frac{1}{4}x = \frac{6}{24}x$
$-\frac{2}{3}x = -\frac{16}{24}x$
$\frac{1}{8}x = \frac{3}{24}x$
Step 3: Combine the terms over the common denominator:
$\left(\frac{6}{24} - \frac{16}{24} + \frac{3}{24}\right)x = 5$
Step 4: Simplify the left side:
$\frac{6 - 16 + 3}{24}x = 5$
$\frac{-7}{24}x = 5$
Step 5: Solve for $x$ by isolating it:
Multiply both sides by $-24/7$ :
$x = 5 \times \left(-\frac{24}{7}\right)$
Step 6: Perform the multiplication:
$x = -\frac{120}{7}$
Step 7: Convert the improper fraction to a mixed number:
The mixed number is $-17\frac{1}{7}$ .

The solution to the equation is $x = -17\frac{1}{7}$ .

Final Answer

$-17\frac{1}{7}$

Key Points to Remember

Essential concepts to master this topic

LCD Method: Find common denominator to combine all fractional coefficients
Technique: Convert $\frac{1}{4}x - \frac{2}{3}x + \frac{1}{8}x$ using LCD = 24
Check: Substitute x = $-17\frac{1}{7}$ back into original equation ✓

Common Mistakes

Avoid these frequent errors

Adding fractions without common denominators
Don't add $\frac{1}{4} - \frac{2}{3} + \frac{1}{8}$ directly = wrong coefficient! This gives incorrect fractions that don't represent the actual x-coefficient. Always find the LCD (24) and convert each fraction first.

Practice Quiz

Test your knowledge with interactive questions

$$ x+7=14 $$

$x=\text{?}$

FAQ

Everything you need to know about this question

Why is the LCD 24 and not 12?

The LCD must work for all denominators: 4, 3, and 8. While 12 works for 4 and 3, it doesn't work for 8 (since 8 doesn't divide evenly into 12). 24 is the smallest number that all three divide into evenly.

How do I convert the improper fraction to a mixed number?

Divide the numerator by the denominator: $-\frac{120}{7}$ means 120 ÷ 7 = 17 remainder 1. So the mixed number is $-17\frac{1}{7}$ .

Why is the answer negative?

The coefficient of x became $\frac{-7}{24}$ (negative), and we're solving $\frac{-7}{24}x = 5$ . When you divide a positive number (5) by a negative fraction, the result must be negative.

Can I solve this without finding the LCD?

You could solve term by term, but it's much more complicated and error-prone. The LCD method is the most efficient way to handle equations with multiple fractions.

How do I check my work with such a complicated fraction?

Substitute $x = -\frac{120}{7}$ back: $\frac{1}{4}(-\frac{120}{7}) - \frac{2}{3}(-\frac{120}{7}) + \frac{1}{8}(-\frac{120}{7})$ . This should equal 5 when simplified.

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