Solve for X: 16(1/4x-2) = 3x+1/2 Linear Equation

Q: How do I convert 65/2 to a mixed number?

Divide 65 by 2: 65 ÷ 2 = 32 remainder 1. So $ \frac{65}{2} = 32\frac{1}{2} $. The quotient becomes the whole number, remainder becomes the new numerator!

Q: What if I made an error in my distribution?

Check each multiplication separately: $ 16 \times \frac{1}{4} = 4 $ and $ 16 \times (-2) = -32 $. Common errors are getting 16x instead of 4x, or forgetting the negative sign.

Q: How do I know when to convert fractions to mixed numbers?

If the answer choices show mixed numbers like $ 32\frac{1}{2} $, convert your improper fraction. Otherwise, either form is mathematically correct!

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 Let's properly open the parentheses, multiply by each factor

00:22 Divide 16 by 4

00:26 Negative times positive is always negative

00:37 Let's isolate the unknown X

00:53 Let's reduce what we can, group the factors

01:06 Let's isolate the unknown 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

1

Understand the problem

Solve for X:

$16\cdot(\frac{1}{4}x-2)=3x+\frac{1}{2}$

2

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Distribute 16 across the expression inside the parentheses.
Step 2: Simplify the equation by moving all terms involving $x$ to one side and constant terms to the other side.
Step 3: Solve for $x$ by isolation through arithmetic manipulation.

Now, let's work through each step:
Step 1: Start by distributing 16 on the left-hand side:

16 \times \left( \frac{1}{4}x - 2 \right) = 16 \times \frac{1}{4}x - 16 \times 2

Simplifying each term gives you:

4x - 32

Now the equation is:

4x - 32 = 3x + \frac{1}{2}

Step 2: Rearrange to collect like terms, move $3x$ to the left side by subtracting it from both sides:

4x - 3x - 32 = \frac{1}{2}

This simplifies to:

x - 32 = \frac{1}{2}

Step 3: Isolate $x$ by adding 32 to both sides:

x = \frac{1}{2} + 32

Converting to a common denominator and performing the addition:

x = \frac{1}{2} + \frac{64}{2}

x = \frac{65}{2}

Finally, convert this fraction to a mixed number:

x = 32\frac{1}{2}

Therefore, the solution to the problem is $x = 32\frac{1}{2}$ .

3

Final Answer

$32\frac{1}{2}$

Key Points to Remember

Essential concepts to master this topic

Distribution: Multiply 16 by every term inside parentheses
Technique: $16 \times \frac{1}{4}x = 4x$ and $16 \times 2 = 32$
Check: Substitute $x = 32\frac{1}{2}$ back into original equation to verify both sides equal ✓

Common Mistakes

Avoid these frequent errors

Forgetting to distribute to all terms
Don't multiply 16 by just $\frac{1}{4}x$ and forget the -2 = incomplete distribution! This leaves -2 instead of -32, giving you the wrong equation to solve. Always multiply the outside number by every single term inside the parentheses.

Practice Quiz

Test your knowledge with interactive questions

$$ x+7=14 $$

$x=\text{?}$

FAQ

Everything you need to know about this question

Why do I get such a large answer like 32½?

+

Large answers are normal in distribution problems! When you distribute 16, it creates bigger coefficients. The -32 from $16 \times (-2)$ is what makes the final answer so large.

How do I convert 65/2 to a mixed number?

+

Divide 65 by 2: 65 ÷ 2 = 32 remainder 1. So $\frac{65}{2} = 32\frac{1}{2}$ . The quotient becomes the whole number, remainder becomes the new numerator!

Can I solve this without distributing first?

+

It's much harder! You'd need to deal with fractions throughout the entire problem. Always distribute first to eliminate parentheses and make the algebra cleaner.

What if I made an error in my distribution?

+

Check each multiplication separately: $16 \times \frac{1}{4} = 4$ and $16 \times (-2) = -32$ . Common errors are getting 16x instead of 4x, or forgetting the negative sign.

How do I know when to convert fractions to mixed numbers?

+

If the answer choices show mixed numbers like $32\frac{1}{2}$ , convert your improper fraction. Otherwise, either form is mathematically correct!

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Solve for X: 16(1/4x-2) = 3x+1/2 Linear Equation

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