Solve the Fraction Equation: Finding X in (6-x)/7 = 3(x-8)/9

Q: When can I use cross-multiplication?

Cross-multiplication works whenever you have one fraction equal to another fraction. The pattern is: if $ \frac{a}{b} = \frac{c}{d} $, then $ a \times d = b \times c $.

Q: What if there are parentheses in the numerator like 3(x-8)?

Treat the entire expression 3(x-8) as one unit when cross-multiplying. So $ \frac{6-x}{7} = \frac{3(x-8)}{9} $ becomes (6-x) × 9 = 3(x-8) × 7. Then expand the parentheses.

Q: Why do I get a decimal answer instead of a whole number?

Many fraction equations have decimal solutions - that's completely normal! When you get $ x = \frac{222}{30} = 7.4 $, this exact value is the correct answer.

Q: How do I check my decimal answer?

Substitute $ x = 7.4 $ back into both sides: $ \frac{6-7.4}{7} = \frac{-1.4}{7} = -0.2 $ and $ \frac{3(7.4-8)}{9} = \frac{3(-0.6)}{9} = -0.2 $. Both sides equal -0.2, so it's correct!

Q: Can I simplify the fraction 222/30 differently?

Yes! $ \frac{222}{30} = \frac{37}{5} = 7.4 $. You can leave it as an improper fraction $ \frac{37}{5} $, mixed number $ 7\frac{2}{5} $, or decimal 7.4 - they're all equivalent!

Cross-Multiplication with Fractional Equations

Solve for X:

$\frac{6-x}{7}=\frac{(x-8)\times3}{9}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Multiply by the common denominator to eliminate fractions

00:22 Simplify as much as possible

00:25 Make sure to open brackets properly, multiply by each factor

01:02 Arrange the equation so that one side has only the unknown X

01:12 Collect like terms

01:18 Isolate X

01:25 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:

$\frac{6-x}{7}=\frac{(x-8)\times3}{9}$

Step-by-step solution

Let's solve the equation $\frac{6-x}{7} = \frac{(x-8)\times3}{9}$ step by step:

Step 1: Cross-multiply to eliminate the fractions:

We have:

$(6-x) \times 9 = ((x-8) \times 3) \times 7$

Step 2: Expand both sides:

Expanding both products gives:

$9(6-x) = 7 \cdot 3(x-8)$

This simplifies to:

$54 - 9x = 21(x-8)$

Step 3: Expand the right side and simplify:

Expanding the right side gives:

$54 - 9x = 21x - 168$

Step 4: Rearrange terms to solve for $x$ :

Bring the terms involving $x$ to one side by adding $9x$ to both sides:

$54 = 30x - 168$

Add 168 to both sides to isolate terms involving $x$ :

$222 = 30x$

Step 5: Solve for $x$ :

Divide both sides by 30 to find $x$ :

$x = \frac{222}{30} = 7.4$

Therefore, the solution to the equation is $x = 7.4$ .

Final Answer

$7.4$

Key Points to Remember

Essential concepts to master this topic

Cross-Multiplication Rule: Multiply diagonally to eliminate both fractions simultaneously
Technique: (6-x) × 9 = 3(x-8) × 7 becomes 54 - 9x = 21x - 168
Check: Substitute x = 7.4: (6-7.4)/7 = 3(7.4-8)/9 gives -0.2 = -0.2 ✓

Common Mistakes

Avoid these frequent errors

Incorrectly cross-multiplying complex fractions
Don't cross-multiply (6-x)/7 = 3(x-8)/9 as (6-x) × 3(x-8) = 7 × 9! This ignores the proper diagonal pattern and creates wrong equations. Always multiply the numerator of each fraction by the denominator of the opposite fraction: (6-x) × 9 = 3(x-8) × 7.

Practice Quiz

Test your knowledge with interactive questions

Solve for X:

$$ 6 - x = 10 - 2 $$

FAQ

Everything you need to know about this question

When can I use cross-multiplication?

Cross-multiplication works whenever you have one fraction equal to another fraction. The pattern is: if $\frac{a}{b} = \frac{c}{d}$ , then $a \times d = b \times c$ .

What if there are parentheses in the numerator like 3(x-8)?

Treat the entire expression 3(x-8) as one unit when cross-multiplying. So $\frac{6-x}{7} = \frac{3(x-8)}{9}$ becomes (6-x) × 9 = 3(x-8) × 7. Then expand the parentheses.

Why do I get a decimal answer instead of a whole number?

Many fraction equations have decimal solutions - that's completely normal! When you get $x = \frac{222}{30} = 7.4$ , this exact value is the correct answer.

How do I check my decimal answer?

Substitute $x = 7.4$ back into both sides: $\frac{6-7.4}{7} = \frac{-1.4}{7} = -0.2$ and $\frac{3(7.4-8)}{9} = \frac{3(-0.6)}{9} = -0.2$ . Both sides equal -0.2, so it's correct!

Can I simplify the fraction 222/30 differently?

Yes! $\frac{222}{30} = \frac{37}{5} = 7.4$ . You can leave it as an improper fraction $\frac{37}{5}$ , mixed number $7\frac{2}{5}$ , or decimal 7.4 - they're all equivalent!

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Solve the Fraction Equation: Finding X in (6-x)/7 = 3(x-8)/9

Cross-Multiplication with Fractional Equations

❤️ 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

When can I use cross-multiplication?

What if there are parentheses in the numerator like 3(x-8)?

Why do I get a decimal answer instead of a whole number?

How do I check my decimal answer?

Can I simplify the fraction 222/30 differently?

🌟 Unlock Your Math Potential

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