91Ó°ÊÓ

The substitution method is a key mathematical technique used to solve equations by replacing variables with their respective expressions. In our exercise, we initially have two equations:

• \( y = \sqrt[3]{5x-1} \)
• \( x = \sqrt{2t+3} \)

To simplify our work, we substitute the expression for x where it appears in the first equation. For example, in step 1, we substitute \( x = \sqrt{2t+3} \) into \( y = \sqrt[3]{5x-1} \). This gives us the new equation: \( y = \sqrt[3]{5\sqrt{2t+3}-1} \). This is a straightforward example of how substitution helps to transform complex equations into a more workable form.

Radical functions include roots, such as square roots or cube roots, in their expressions. In this exercise, both the original equations for y and x are radical functions. The function \( y = \sqrt[3]{5x-1} \) is a cube root function, while \( x = \sqrt{2t+3} \) is a square root function. Radical functions often involve unique operations due to the nature of taking roots. They can be challenging but remembering to work step-by-step can simplify the process. In particular, when dealing with expressions like \( \sqrt{a+b} \) and \( \sqrt[3]{a+b} \), be careful to perform all steps accurately.

A cube root function is a type of radical function where we determine the number that, when multiplied by itself three times, yields the given number. In our case, the function \( y = \sqrt[3]{5x-1} \) involves taking the cube root of the expression \( 5x-1 \). Note that cube roots can handle both positive and negative values, unlike square roots which are generally non-negative. The cube root function is important because it allows us to solve for y in terms of x, even when x itself is defined in terms of another variable, like t.To solve for y when x is given as \( \sqrt{2t+3} \), we first substitute this into the function, yielding: \( y = \sqrt[3]{5\sqrt{2t+3}-1} \). Understanding cube roots will help you grasp more complex mathematical functions and their behaviors.