Solve for X: Finding X in -1/4x + 8 = 3x - 1/2

Q: Why did we get a mixed number answer instead of a whole number?

Not all equations have whole number solutions! $ x = 2\frac{8}{13} $ is the exact answer. Mixed numbers are perfectly valid solutions in algebra.

Q: How do I multiply every term by 4 without making mistakes?

Go term by term: $ 4 \times (-\frac{1}{4}x) = -x $, $ 4 \times 8 = 32 $, $ 4 \times 3x = 12x $, and $ 4 \times (-\frac{1}{2}) = -2 $. Check each multiplication separately!

Q: What if I can't convert the improper fraction to a mixed number?

To convert $ \frac{34}{13} $: divide 34 ÷ 13 = 2 remainder 8. So $ \frac{34}{13} = 2\frac{8}{13} $. The quotient becomes the whole number, remainder becomes the new numerator.

Q: How can I check if my mixed number answer is correct?

Convert $ 2\frac{8}{13} $ back to improper fraction: $ \frac{34}{13} $. Then substitute into the original equation. Both sides should equal the same decimal value when calculated.

Q: Is there a faster way to solve this without clearing fractions?

You could work with fractions throughout, but it's much more error-prone. The LCD method makes the algebra cleaner and reduces calculation mistakes significantly.

Linear Equations with Mixed Number Solutions

Solve for X:

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's solve the problem.

00:10 First, we need to isolate the unknown X. Let's get started!

00:29 Next, reduce what we can to make the equation simpler.

00:42 Now, collect the like terms together for clarity.

00:53 Our goal is to isolate X. Let's focus on that!

01:09 Once again, reduce what you can for simplicity.

01:27 Multiply by X's reciprocal fraction to isolate it.

01:44 Remember, multiply numerator by numerator and denominator by denominator, just like before.

01:57 Great job! And that's the solution to our question.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve for X:

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

Step-by-step solution

To solve the equation $-\frac{1}{4}x + 8 = 3x - \frac{1}{2}$ , follow these steps:

Step 1: Remove fractions by multiplying every term by the least common multiple of the denominators, which is 4.
This gives $4 \left(-\frac{1}{4}x\right) + 4 \times 8 = 4 \times 3x - 4 \times \frac{1}{2}$ , simplifying to $-x + 32 = 12x - 2$ .
Step 2: Move all terms involving $x$ to one side by adding $x$ to both sides.
This gives $32 = 13x - 2$ .
Step 3: Isolate $x$ by moving the constant to the other side: Add 2 to both sides to obtain $32 + 2 = 13x$ , simplifying to $34 = 13x$ .
Step 4: Solve for $x$ by dividing both sides by 13, leading to $x = \frac{34}{13}$ .
Step 5: Simplify the fraction if possible. Here, $\frac{34}{13}$ is already in its simplest form. Converting to a mixed number, we get $2\frac{8}{13}$ .

Therefore, the solution to the equation is $x = 2\frac{8}{13}$ .

Final Answer

$2\frac{8}{13}$

Key Points to Remember

Essential concepts to master this topic

LCD Method: Multiply every term by 4 to eliminate fractions completely
Collect Terms: Move all x terms to one side: -x + x becomes 0x
Verify Answer: Substitute $x = 2\frac{8}{13}$ back into original equation ✓

Common Mistakes

Avoid these frequent errors

Forgetting to multiply all terms by the LCD
Don't multiply only $-\frac{1}{4}x$ by 4 and leave 8 unchanged = unbalanced equation giving wrong answer! This destroys the equality because you're not doing the same operation to both sides. Always multiply every single term on both sides by the LCD.

Practice Quiz

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$$ 11=a-16 $$

$a=\text{?}$

FAQ

Everything you need to know about this question

Why did we get a mixed number answer instead of a whole number?

Not all equations have whole number solutions! $x = 2\frac{8}{13}$ is the exact answer. Mixed numbers are perfectly valid solutions in algebra.

How do I multiply every term by 4 without making mistakes?

Go term by term: $4 \times (-\frac{1}{4}x) = -x$ , $4 \times 8 = 32$ , $4 \times 3x = 12x$ , and $4 \times (-\frac{1}{2}) = -2$ . Check each multiplication separately!

What if I can't convert the improper fraction to a mixed number?

To convert $\frac{34}{13}$ : divide 34 ÷ 13 = 2 remainder 8. So $\frac{34}{13} = 2\frac{8}{13}$ . The quotient becomes the whole number, remainder becomes the new numerator.

How can I check if my mixed number answer is correct?

Convert $2\frac{8}{13}$ back to improper fraction: $\frac{34}{13}$ . Then substitute into the original equation. Both sides should equal the same decimal value when calculated.

Is there a faster way to solve this without clearing fractions?

You could work with fractions throughout, but it's much more error-prone. The LCD method makes the algebra cleaner and reduces calculation mistakes significantly.

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