Solve for X: Finding X in ½(8-x) + ½x = 12-x Linear Equation

Q: Why do the x terms cancel out on the left side?

After distributing, you get $ -\frac{1}{2}x + \frac{1}{2}x $. These are opposite terms that add to zero! This simplifies the equation significantly.

Q: What if I don't see that the x terms cancel?

No problem! You can still solve by collecting like terms. Combine the x terms: $ -\frac{1}{2}x + \frac{1}{2}x = 0x = 0 $, then continue solving.

Q: Can I multiply both sides by 2 to clear fractions first?

Yes! Multiplying by 2 gives: $ (8-x) + x = 24 - 2x $. This also works, but you'll still need to use the distributive property afterward.

Q: How do I check if x = 8 is correct?

Substitute into the original equation:Left side: $ \frac{1}{2}(8-8) + \frac{1}{2}(8) = 0 + 4 = 4 $Right side: $ 12 - 8 = 4 $Since both sides equal 4, the answer is correct!

Q: What if I get a different answer?

Double-check your distribution step and make sure you correctly combined like terms. The most common error is sign mistakes when distributing negative fractions.

Linear Equations with Distributive Property Simplification

Solve for X:

$\frac{1}{2}\cdot(8-x)+\frac{1}{2}x=12-x$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:10 Let's solve this together!

00:14 First, we want to isolate the unknown variable X.

00:18 Make sure to open the parentheses correctly, and multiply each term inside.

00:34 Next, divide 8 by 2.

00:37 Remember, a positive times a negative gives a negative.

00:48 Now, let's simplify whatever we can.

00:54 Our goal is to isolate the unknown, X.

01:10 Again, let's simplify the expression.

01:20 Multiply by negative 1 to change a negative to a positive.

01:30 And remember, negative times negative gives us a positive.

01:35 And there you have it! That's the solution to our problem.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve for X:

$\frac{1}{2}\cdot(8-x)+\frac{1}{2}x=12-x$

Step-by-step solution

To solve the equation $\frac{1}{2} \cdot (8-x) + \frac{1}{2}x = 12 - x$ , follow these steps:

Step 1: Use the distributive property on the left side of the equation.
- Distribute $\frac{1}{2}$ across $(8-x)$ :
- This gives us $\frac{1}{2} \cdot 8 - \frac{1}{2} \cdot x + \frac{1}{2}x$ .
- Simplify the terms: $4 - \frac{1}{2}x + \frac{1}{2}x$ .
Step 2: Simplify the left side of the equation:
- The terms $-\frac{1}{2}x$ and $\frac{1}{2}x$ cancel each other out.
- Thus, the left side simplifies to $4$ .
Step 3: Set the simplified equation equal to the right side:
- So, we have $4 = 12 - x$ .
Step 4: Solve for $x$ using basic algebra:
- Add $x$ to both sides to move $x$ to one side: $x + 4 = 12$ .
- Subtract 4 from both sides to solve for $x$ : $x = 12 - 4$ .
- This results in $x = 8$ .

Therefore, the solution to the equation is $x = 8$ .

Final Answer

$8$

Key Points to Remember

Essential concepts to master this topic

Distributive Property: Multiply each term inside parentheses by the coefficient
Like Terms: Combine $-\frac{1}{2}x + \frac{1}{2}x = 0$ to simplify left side
Verification: Substitute x = 8: $\frac{1}{2}(0) + 4 = 4$ and $12 - 8 = 4$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to distribute the fraction to all terms
Don't just multiply $\frac{1}{2}$ by 8 and ignore the x term = $4 + \frac{1}{2}x$ instead of $4 - \frac{1}{2}x$ ! This changes the signs and gives wrong answers. Always distribute the coefficient to 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 the x terms cancel out on the left side?

After distributing, you get $-\frac{1}{2}x + \frac{1}{2}x$ . These are opposite terms that add to zero! This simplifies the equation significantly.

What if I don't see that the x terms cancel?

No problem! You can still solve by collecting like terms. Combine the x terms: $-\frac{1}{2}x + \frac{1}{2}x = 0x = 0$ , then continue solving.

Can I multiply both sides by 2 to clear fractions first?

Yes! Multiplying by 2 gives: $(8-x) + x = 24 - 2x$ . This also works, but you'll still need to use the distributive property afterward.

How do I check if x = 8 is correct?

Substitute into the original equation:

Left side: $\frac{1}{2}(8-8) + \frac{1}{2}(8) = 0 + 4 = 4$
Right side: $12 - 8 = 4$

Since both sides equal 4, the answer is correct!

What if I get a different answer?

Double-check your distribution step and make sure you correctly combined like terms. The most common error is sign mistakes when distributing negative fractions.

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