Find the Domain -x+5y=10
The question is asking you to determine the set of all possible values that the variable 'y' can take, while 'x' can be any real number, in the linear equation -x+5y=10. The domain, in this context, is related to the independent variable, which is typically 'x', and you need to identify the values of 'x' for which this equation is valid. However, in the context of functions, finding the domain usually means identifying all the allowable 'x' values for which the function (relationship between 'x' and 'y') is defined. But since this is a linear equation and not explicitly a function, and because 'x' typically represents the independent variable, there may be some confusion in the phrasing of the question.
$- x + 5 y = 10$
Answer
Solution:
Step 1:
Isolate $y$ by moving $x$ to the other side of the equation: $5y = x + 10$.
Step 2:
Divide the equation $5y = x + 10$ by $5$ to solve for $y$.
Step 2.1:
Apply division to each term: $\frac{5y}{5} = \frac{x}{5} + \frac{10}{5}$.
Step 2.2:
Reduce the fraction on the left-hand side.
Step 2.2.1:
Eliminate the common factor of $5$: $\frac{\cancel{5}y}{\cancel{5}} = \frac{x}{5} + \frac{10}{5}$.
Step 2.2.1.1:
Simplify $y$: $y = \frac{x}{5} + \frac{10}{5}$.
Step 2.3:
Simplify the constant term on the right-hand side.
Step 2.3.1:
Compute $\frac{10}{5}$: $y = \frac{x}{5} + 2$.
Step 3:
Determine the domain of $y$. Since $y$ is defined for all real values of $x$, the domain is all real numbers.
Interval Notation: $(-\infty, \infty)$
Set-Builder Notation: $\{x | x \in \mathbb{R}\}$
Step 4:
There is no additional step required as the domain has been established.
Knowledge Notes:
The domain of a function or expression is the set of all possible input values (usually represented by $x$) for which the function or expression is defined. To find the domain of a linear equation like $-x + 5y = 10$, we solve for $y$ in terms of $x$ and then determine if there are any restrictions on the values that $x$ can take.
In this case, the equation is linear and does not contain any terms that would restrict the domain, such as square roots or denominators that could become zero. Therefore, the domain is all real numbers, denoted by $\mathbb{R}$.
Interval notation is a way of writing subsets of the real number line. An interval that includes all real numbers is written as $(-\infty, \infty)$.
Set-builder notation is another way to describe a set, defining the properties that its members must satisfy. For the domain of all real numbers, it is written as $\{x | x \in \mathbb{R}\}$, which reads as "the set of all $x$ such that $x$ is an element of the real numbers."
When simplifying algebraic expressions, common factors in numerators and denominators can be canceled. For example, $\frac{5y}{5}$ simplifies to $y$ because the common factor of $5$ is divided out.
In summary, the domain of the given linear equation is all real numbers because there are no algebraic restrictions on the value of $x$.
