Solve the Fraction Equation: Finding X in (1/3)(x+9) = 4 + (2/3)x

Linear Equations with Fractional Coefficients

13(x+9)=4+23x \frac{1}{3}(x+9)=4+\frac{2}{3}x

x=? x=\text{?}

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Solve
00:04 Multiply by the denominator to eliminate the fraction
00:23 Simplify what we can
00:30 We want to isolate the unknown X
00:34 Arrange the equation so that X is alone on one side
00:51 Convert from negative to positive
00:58 Negative divided by negative always equals positive
01:01 Positive divided by negative always equals negative
01:04 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

13(x+9)=4+23x \frac{1}{3}(x+9)=4+\frac{2}{3}x

x=? x=\text{?}

2

Step-by-step solution

To solve the equation 13(x+9)=4+23x \frac{1}{3}(x+9) = 4+\frac{2}{3}x , we will follow these steps:

  • Step 1: Clear fractions by multiplying through by the least common denominator.
  • Step 2: Simplify the equation to combine like terms.
  • Step 3: Solve for x x .

Let's begin:

Step 1: Multiply every term in the equation by 3 to eliminate fractions:

3⋅13(x+9)=3⋅(4+23x) 3 \cdot \frac{1}{3}(x+9) = 3 \cdot \left( 4 + \frac{2}{3}x \right)

This simplifies to:

x+9=12+2x x + 9 = 12 + 2x

Step 2: Rearrange the equation to get all x x terms on one side and constant terms on the other:

Subtract 2x 2x from both sides:

x+9−2x=12 x + 9 - 2x = 12

Which simplifies to:

−x+9=12 -x + 9 = 12

Next, subtract 9 from both sides to isolate terms involving x x :

−x=3 -x = 3

Step 3: Solve for x x by multiplying both sides by -1:

x=−3 x = -3

Thus, the solution to the equation is x=−3 x = -3 .

3

Final Answer

3-

Key Points to Remember

Essential concepts to master this topic
  • Rule: Multiply both sides by LCD to eliminate all fractions
  • Technique: For LCD 3, multiply each term: 3⋅13(x+9)=3⋅4 3 \cdot \frac{1}{3}(x+9) = 3 \cdot 4
  • Check: Substitute x = -3: 13(−3+9)=4+23(−3) \frac{1}{3}(-3+9) = 4 + \frac{2}{3}(-3) gives 2 = 2 ✓

Common Mistakes

Avoid these frequent errors
  • Only multiplying some terms by the LCD
    Don't multiply just 13(x+9) \frac{1}{3}(x+9) by 3 and leave other terms unchanged = unbalanced equation! This destroys the equality and gives wrong answers. Always multiply every single term on both sides by the LCD.

Practice Quiz

Test your knowledge with interactive questions

\( x+7=14 \)

\( x=\text{?} \)

FAQ

Everything you need to know about this question

Why multiply everything by 3 instead of just the fractions?

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You must multiply every term on both sides to keep the equation balanced! If you only multiply some terms, you're changing the equation completely and will get the wrong answer.

How do I know what LCD to use?

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Look at all the denominators in your equation. Here we have 13 \frac{1}{3} and 23 \frac{2}{3} , so the LCD is 3. For different denominators like 2 and 4, the LCD would be 4.

What if I get a negative answer like x = -3?

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Negative answers are completely normal! Many equations have negative solutions. Just make sure to substitute carefully when checking: negative times negative equals positive.

Can I solve this without clearing fractions first?

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Yes, but it's much harder! You'd have to work with fractions throughout. Clearing fractions first makes the algebra much simpler and reduces calculation errors.

How do I check my answer when there are fractions?

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Substitute x = -3 into the original equation: 13(−3+9)=13(6)=2 \frac{1}{3}(-3+9) = \frac{1}{3}(6) = 2 and 4+23(−3)=4−2=2 4 + \frac{2}{3}(-3) = 4 - 2 = 2 . Both sides equal 2, so it's correct!

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