How to Solve for y in Terms of x: A Comprehensive Guide

Understanding how to solve for y in terms of x is a fundamental skill in algebra. This guide walks through various methods and examples to help you master the process.

Introduction to Solving for y in Terms of x

When you see an equation involving both y and x, the goal is to rearrange the equation so that y is the subject. This means that the equation should be written as y some expression involving x. This process is also known as isolating y.

General Steps to Solve for y in Terms of x

Rearrange the Equation: Move all terms involving y to one side of the equation, and move all other terms to the other side. Algebraic Operations: Use addition, subtraction, multiplication, and division to isolate y. Simplify the Expression: Simplify the final expression if possible.

Example: Solving x^2y 5

Let us solve for y in the equation x^2y 5.

x^2y 5 y 5 - x^2

In this case, y is already isolated. The equation is now in the form y 5 - x^2.

Example: Solving for y in

Now let's consider a more complex equation: .

Let z e^y. The equation becomes . Simplify the expression: frac{1}{2} x. Solve for z: becomes , hence z 2x. Since z e^y, we have e^y 2x. Take the natural logarithm of both sides: y ln(2x). This equation can also be represented in terms of a hyperbolic function, such as: x cosh(y), where y arcosh(x).

Note: cosh(x) is the hyperbolic cosine function, and arcosh(x) is the inverse hyperbolic cosine function.

Conclusion

Solving for y in terms of x requires a step-by-step approach, utilizing basic algebraic operations and in some cases, properties of exponential and hyperbolic functions. By following these steps, you can isolate y and express it in terms of x.

Key Points to Remember:

Move all terms involving y to one side of the equation. Utilize algebraic operations to isolate y. Simplify the expression to its final form.