Solving Equations Using All Methods: Domain of definition

To determine the field of application of the equation $\frac{3x:4}{y+6}=6$, we must identify values of $y$ for which the equation is defined.

• The denominator of the given expression is $y + 6$. In order for the expression to be defined, the denominator cannot be zero.
• This leads us to solve the equation $y + 6 = 0$.
• Solving $y + 6 = 0$ gives us $y = -6$.
• This means $y = -6$ would make the denominator zero, thus the expression would be undefined for this value.

Therefore, the field of application, or the domain of the equation, is all real numbers except $y = -6$.

We must conclude that $y \neq -6$.

Comparing with the provided choices, the correct answer is choice 3: $y \neq -6$.

To solve the problem, follow these steps:

• Step 1: Understand that the equation $\frac{25a+4b}{7y + 4 \cdot 3 + 2}=9b$ is undefined when the denominator equals zero.
• Step 2: Simplify the denominator: $7y + 4 \cdot 3 + 2$.
• Step 3: Calculate the constant part: $4 \cdot 3 = 12$, so the expression becomes $7y + 12 + 2$.
• Step 4: Combine constants: $12 + 2 = 14$. The denominator is $7y + 14$.
• Step 5: Set the denominator equal to zero to find values of $y$ that make the equation undefined: $7y + 14 = 0$.
• Step 6: Solve for $y$:
• Subtract 14 from both sides: $7y = -14$.
• Divide by 7: $y = -2$.

Therefore, the equation is undefined when $y = -2$. The field of application excludes $y = -2$.

The choice that reflects this is $\boxed{y \neq -2}$.

To solve this problem, we'll follow these steps to find the domain:

• Step 1: Recognize that the expression $\frac{x+y:3}{2x+6}=4$ involves a fraction. The denominator $2x + 6$ must not be zero, as division by zero is undefined.
• Step 2: Set the denominator equal to zero and solve for $x$ to find the values that must be excluded: $2x + 6 = 0$.
• Step 3: Solve $2x + 6 = 0$:
• $2x + 6 = 0$
• $2x = -6$
• $x = -3$
• Step 4: Conclude that the domain of the function excludes $x = -3$, meaning $x \neq -3$.

Thus, the domain of the given expression is all real numbers except $x = -3$. This translates to:

$x\operatorname{\ne}-3$

To solve this problem, we will determine the domain, or field of application, of the equation $\frac{6}{x+5} = 1$.

Step-by-step solution:

• Step 1: Identify the denominator. In the given equation, the denominator is $x+5$.
• Step 2: Determine when the denominator is zero. Solve for $x$ by setting $x+5 = 0$.
• Step 3: Solve the equation: $x+5 = 0$ gives $x = -5$.
• Step 4: Exclude this value from the domain. The domain is all real numbers except $x = -5$.

Therefore, the field of application of the equation is all real numbers except where $x = -5$.

Thus, the domain is $x \neq -5$.

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: $y \neq -5$.

To determine the field of application (domain of definition) of the given equation, we need to identify all values of the variables that would make the equation undefined.

The given equation is:
$\frac{\sqrt{15}+34:z}{4y-12+8:2}=5$

where the colon (:) represents division, so we can rewrite this as:
$\frac{\sqrt{15}+\frac{34}{z}}{4y-12+\frac{8}{2}}=5$

Step 1: Identify potential division by zero in the numerator
In the numerator, we have the term $\frac{34}{z}$. This expression is undefined when $z = 0$.
Therefore, we must have: $z \neq 0$

Step 2: Simplify and analyze the denominator
The denominator is: $4y-12+\frac{8}{2}$
Simplifying: $4y-12+4 = 4y-8$

Step 3: Identify when the denominator equals zero
The entire fraction is undefined when the denominator equals zero:
$4y-8 = 0$
$4y = 8$
$y = 2$

Therefore, we must have: $y \neq 2$

Step 4: State the domain (field of application)
The equation is defined for all values of $y$ and $z$ except those that cause division by zero.

Therefore, the field of application of the equation is: $z \neq 0$ and $y \neq 2$

This corresponds to choice 3.

To find the domain of definition for this equation, we need to identify all values of the variables that would make the expression undefined. An expression becomes undefined when we have division by zero.

Let me first clarify the notation and rewrite the equation properly:
$\frac{\frac{24}{3t}+4}{24y+21\cdot2+6}=7z$

Now, let's identify where division by zero could occur:

Step 1: Analyze the numerator
The numerator is $\frac{24}{3t}+4$. For this to be defined, we need:
$3t \neq 0$
Therefore: $t \neq 0$

Step 2: Analyze the denominator
The denominator is $24y+21\cdot2+6$. Let's simplify this:
$24y + 42 + 6 = 24y + 48$

For the entire fraction to be defined, we need:
$24y + 48 \neq 0$
$24y \neq -48$
$y \neq -2$

Step 3: State the domain restrictions
For the equation to be defined, both conditions must be satisfied simultaneously:

• $t \neq 0$ (to avoid division by zero in the numerator)
• $y \neq -2$ (to avoid division by zero in the main denominator)

Therefore, the domain of definition is: $y \neq -2, t \neq 0$