Solving Equations Using All Methods: Domain of definition

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 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 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 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 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 solve this problem, we need to identify the values of $y$ for which the denominator of the expression becomes zero, as these values are not part of the domain.

First, let's simplify the denominator of the given equation:

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

Simplifying the terms: $34:z \text{ remains as it is for simplification purposes, and } \frac{8}{2} = 4$

Thus, the denominator becomes: $(4y - 12 + 4) = 4y - 8$

We need to ensure the denominator is not zero to avoid undefined expressions: $4y - 8 \neq 0$

Simplify and solve for $y$: $4y - 8 \neq 0 \implies 4y \neq 8 \implies y \neq 2$

Therefore, the equation is undefined for $y = 2$, and the answer is that the field of application excludes $y = 2$.

Given the possible choices for the problem, the correct choice is: $y\operatorname{\ne}2$

The solution to this problem is $y \neq 2$.