Let's analyze each equation to determine if it specifies a function with an independent variable $x$, and if so, find the domain of the function. If it doesn't specify a function, we'll look for values of $x$ that correspond to more than one value of $y$.

a) $2x + 5y = 10$

To determine if this equation represents a function of $x$, we solve for $y$ in terms of $x$:

$5y = 10 - 2x$ $y = \frac{10 - 2x}{5}$

This is a linear equation, so it specifies $y$ as a function of $x$. The domain of this function is all real numbers, since there are no restrictions on $x$.

Domain: $(-\infty, \infty)$

b) $y(x + y) = 4$

First, let's try to isolate $y$ to see if it represents $y$ as a function of $x$:

$y(x + y) = 4$

Expanding the left-hand side:

$yx + y^2 = 4$

This is a quadratic equation in $y$ (i.e., $y^2 + xy - 4 = 0$). Solving this quadratic for $y$ using the quadratic formula:

$y = \frac{-x \pm \sqrt{x^2 + 16}}{2}$

Since there is a "±" sign in the solution, this means that for certain values of $x$, there are two possible values of $y$. Thus, this equation does not define $y$ as a function of $x$ because for some values of $x$, there are multiple values of $y$.

For instance, if $x = 0$:

$y = \frac{-0 \pm \sqrt{0^2 + 16}}{2} = \frac{\pm 4}{2} = \pm 2$

So for $x = 0$, there are two values of $y$ ($y = 2$ and $y = -2$).

c) $x(x + y) = 4$

Let's try to solve for $y$ in terms of $x$:

$x(x + y) = 4$ $x^2 + xy = 4$ $xy = 4 - x^2$ $y = \frac{4 - x^2}{x}$

This expression for $y$ is defined as long as $x \neq 0$ (since division by zero is undefined). So, this equation does define $y$ as a function of $x$, but with the restriction that $x \neq 0$.

Domain: $(-\infty, 0) \cup (0, \infty)$

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Summary:

• a) $2x + 5y = 10$ defines $y$ as a function of $x$. Domain: $(-\infty, \infty)$
• b) $y(x + y) = 4$ does not define $y$ as a function of $x$. Example: $x = 0$ gives two values of $y$.
• c) $x(x + y) = 4$ defines $y$ as a function of $x$ with the domain $x \neq 0$, or $(-\infty, 0) \cup (0, \infty)$.

Do you want more details, or do you have any questions?

Related Questions:

1. How do you determine if an equation represents a function?
2. What is the quadratic formula and how is it used to solve quadratic equations?
3. How do you find the domain of a function involving a fraction?
4. Why can't the domain include values where division by zero occurs?
5. What are the key differences between linear and quadratic equations?

Tip: When determining if an equation represents a function, always check if each value of the independent variable corresponds to exactly one value of the dependent variable.