Solve each equation for the specified variable. (Assume all denominators are nonzero.) x^2/3+y^2/3=a^2/3, for y

Verified step by step guidance

1. Start with the given equation: \(\frac{x^{2}}{3} + \frac{y^{2}}{3} = \frac{a^{2}}{3}\).
2. To isolate the term with \(y\), subtract \(\frac{x^{2}}{3}\) from both sides: \(\frac{y^{2}}{3} = \frac{a^{2}}{3} - \frac{x^{2}}{3}\).
3. Combine the right side over a common denominator: \(\frac{y^{2}}{3} = \frac{a^{2} - x^{2}}{3}\).
4. Multiply both sides of the equation by 3 to eliminate the denominator: \(y^{2} = a^{2} - x^{2}\).
5. Take the square root of both sides to solve for \(y\): \(y = \pm \sqrt{a^{2} - x^{2}}\).

Key Concepts

Isolating a Variable

Isolating a variable means rewriting an equation so that the variable of interest stands alone on one side. This often involves algebraic operations like addition, subtraction, multiplication, division, and taking roots to express the variable explicitly.

Fractional Exponents

Fractional exponents represent roots and powers simultaneously. For example, an exponent of 2/3 means taking the cube root first and then squaring, or vice versa. Understanding how to manipulate fractional exponents is key to solving equations involving them.

Domain Restrictions and Denominators

When solving equations with denominators, it is important to consider domain restrictions to avoid division by zero. Ensuring denominators are nonzero maintains the equation's validity and helps identify permissible values for variables.