Solving xyz/(2(3+y)+4) = 8: Understanding the Field of Application

Q: What exactly is the 'field of application' or domain?

The domain (or field of application) is all the values that make the expression valid. For fractions, we exclude any values that make the denominator zero because division by zero is undefined.

Q: Why can't the denominator be zero?

Division by zero is undefined in mathematics! When the denominator equals zero, the fraction has no meaning. That's why we must exclude these values from the domain.

Q: How do I solve 2(3+y)+4 = 0 step by step?

Step 1: Distribute: $ 6 + 2y + 4 = 0 $Step 2: Combine like terms: $ 2y + 10 = 0 $Step 3: Subtract 10: $ 2y = -10 $Step 4: Divide by 2: $ y = -5 $

Q: What's the difference between y = -5 and y ≠ -5?

$ y = -5 $ means y equals -5, but $ y \neq -5 $ means y can be any value except -5. Since -5 makes our denominator zero, we exclude it from the domain.

Domain Restrictions with Rational Expressions

$\frac{xyz}{2(3+y)+4}=8$

What is the field of application of the equation?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the domain of the function

00:03 According to mathematical laws, division by 0 is forbidden

00:07 Since there is a variable in the denominator, we must ensure it is not 0

00:15 To do this, set the denominator equal to 0 and solve

00:18 Properly expand the parentheses, multiply by each factor

00:30 Collect like terms

00:38 Isolate the variable Y

00:52 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$\frac{xyz}{2(3+y)+4}=8$

What is the field of application of the equation?

Step-by-step solution

To find the domain of the given equation $\frac{xyz}{2(3+y)+4}=8$ , we need to ensure the denominator is not zero. This means solving $2(3+y) + 4 = 0$ .

Let's solve this step-by-step:

Step 1: Simplify the expression $2(3+y) + 4 = 0$ .
Step 2: Expand to $6 + 2y + 4 = 0$ .
Step 3: Combine like terms to get $2y + 10 = 0$ .
Step 4: Isolate the variable $y$ . Subtract 10 from both sides: $2y = -10$ .
Step 5: Divide by 2 to solve for $y$ : $y = -5$ .

If $y = -5$ , the denominator becomes zero, which makes the original expression undefined.

Therefore, the value of $y$ must not be $-5$ for the expression to be valid. In conclusion, the restriction on $y$ is that $y \neq -5$ .

The correct answer choice is: $y \neq -5$ .

Final Answer

$y\ne-5$

Key Points to Remember

Essential concepts to master this topic

Domain Rule: Denominator cannot equal zero in any rational expression
Technique: Set denominator equal to zero: $2(3+y)+4 = 0$
Check: Verify $y = -5$ makes denominator zero, so $y \neq -5$ ✓

Common Mistakes

Avoid these frequent errors

Ignoring the denominator when finding domain
Don't just look at the numerator or focus only on solving the equation = missing critical restrictions! The denominator determines where the expression is undefined. Always identify values that make the denominator zero first.

Practice Quiz

Test your knowledge with interactive questions

Solve for X:

$$ 6 - x = 10 - 2 $$

FAQ

Everything you need to know about this question

What exactly is the 'field of application' or domain?

The domain (or field of application) is all the values that make the expression valid. For fractions, we exclude any values that make the denominator zero because division by zero is undefined.

Why can't the denominator be zero?

Division by zero is undefined in mathematics! When the denominator equals zero, the fraction has no meaning. That's why we must exclude these values from the domain.

Do I need to worry about the numerator being zero?

No! If the numerator equals zero, the fraction just equals zero, which is perfectly fine. Only worry about the denominator when finding domain restrictions.

How do I solve 2(3+y)+4 = 0 step by step?

Step 1: Distribute: $6 + 2y + 4 = 0$

Step 2: Combine like terms: $2y + 10 = 0$

Step 3: Subtract 10: $2y = -10$

Step 4: Divide by 2: $y = -5$

What's the difference between y = -5 and y ≠ -5?

$y = -5$ means y equals -5, but $y \neq -5$ means y can be any value except -5. Since -5 makes our denominator zero, we exclude it from the domain.

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Solving xyz/(2(3+y)+4) = 8: Understanding the Field of Application

Domain Restrictions with Rational Expressions

❤️ 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

What exactly is the 'field of application' or domain?

Why can't the denominator be zero?

Do I need to worry about the numerator being zero?

How do I solve 2(3+y)+4 = 0 step by step?

What's the difference between y = -5 and y ≠ -5?

🌟 Unlock Your Math Potential

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