Solving for (y) in Terms of (x): Step-by-Step Guide with Examples
Solving for (y) in terms of (x) is a fundamental algebraic skill that can be applied in numerous mathematical contexts. This guide will walk you through the process with clear examples and step-by-step instructions to help you understand how to isolate (y) in equations. Whether you are a student, a teacher, or someone who simply enjoys solving math problems, this guide will be a valuable resource.
Introduction to Solving for (y)
Solving for (y) in terms of (x) means we want to express (y) as a function of (x). Essentially, we need to isolate (y) on one side of the equation, so that it only involves (x) and possibly constants.
Step-by-Step Process
Here is a step-by-step guide for solving for (y) in a given equation:
Identify the equation: Start with the given equation. For example, consider the equation (2y - 3x 12). Isolate (y): Perform algebraic operations to get (y) by itself on one side of the equation. Subtract (3x) from both sides to get:2y 12 - 3x
Simplify the expression: To make the expression more concise, we can divide both sides by 2:y (frac{12 - 3x}{2})
Further simplification (if possible): The expression can be simplified even further:y 6 - (frac{3}{2}x)
Now, (y) is expressed in terms of (x).
Examples and Applications
Let's explore some examples to solidify your understanding.
Example 1: (2y - 3x 12)
As illustrated above, the steps are as follows:
Start with the equation: (2y - 3x 12). Subtract (3x) from both sides: (2y 12 - 3x). Divide both sides by 2: (y frac{12 - 3x}{2}). Simplify: (y 6 - frac{3}{2}x).Example 2: (xy 5)
To solve for (y), follow these steps:
Multiply both sides by (frac{1}{x}): (y frac{5}{x}).This gives us (y) in terms of (x).
Example 3: (frac{e^y e^{-y}}{2} x)
Here’s a more complex example:
Let (z e^y). Substitute into the equation: (frac{z cdot frac{1}{z}}{2} x). Simplify: (frac{z}{2z} x). This reduces to (frac{1}{2} x), but this is not the correct simplification. Let's solve for (z) first: Recall that (frac{z cdot frac{1}{z}}{2} x) simplifies to (frac{1}{2} x), but we need to isolate (z): (z x cdot 2z).Correctly, we should solve the quadratic equation:
(frac{z^2}{2} - xz frac{1}{2} 0)
Solving the quadratic equation using the quadratic formula (z frac{-b pm sqrt{b^2 - 4ac}}{2a}), we get:
(z x pm sqrt{x^2 - 1})
Since (z e^y), we take the natural logarithm:
(y ln(x pm sqrt{x^2 - 1}))
This can also be expressed using the hyperbolic cosine function:
(y text{arccosh}(x))
Conclusion
Solving for (y) in terms of (x) is a key skill in algebra and many real-world applications. By following the step-by-step process, you can tackle even the most complex equations. Remember to isolate (y) and simplify where possible. With practice, you will become more proficient in solving for variables in various equations.