Problem 71 - Algebra for College Students

Question

Find the indicated value for each given rational expression, if possible. $$H(y)=\frac{y^{2}-5}{3 y-4}, H(-2)$$

Step-by-Step Solution

Answer

H(-2) = \frac{1}{10}.

Step 1: Understand the problem

The task requires evaluating the rational function \(H(y)\) at \(y = -2\). The given rational expression is \[H(y) = \frac{y^{2}-5}{3y-4}\].

Step 2: Substitute the value of y

Substitute \(-2\) for \(y\) in the rational expression. So, \[H(-2) = \frac{(-2)^{2}-5}{3(-2)-4}\].

Step 3: Simplify the numerator

Calculate the numerator by simplifying \((-2)^{2}-5\). So, \((-2)^{2} = 4\) and \(4 - 5 = -1\). The numerator is \(-1\).

Step 4: Simplify the denominator

Calculate the denominator by simplifying \(3(-2)-4\). So, \(3(-2) = -6\) and \(-6 - 4 = -10\). The denominator is \(-10\).

Step 5: Form the rational expression

Now, form the rational expression with the simplified numerator and denominator. So, \[H(-2) = \frac{-1}{-10} = \frac{1}{10}\].

Step 6: Check the solution

Verify that substituting \(-2\) does not cause the denominator to be zero. Since the denominator is \(-10\), the expression is valid.

Key Concepts

rational functions

A rational function is a ratio of two polynomial expressions. Consider the rational function from the problem:



Rational functions can behave differently depending on their inputs. The most important thing to remember is that the denominator should never be zero, as this makes the expression undefined. When evaluating a given rational function at a certain point, always start by checking that the substitution point doesn't make the denominator zero.

substitution method

The substitution method is a key technique for evaluating expressions, including rational functions. Here's a simple way to break it down:

• Identify the value that needs to be substituted into the expression.
  
• Replace every occurrence of the variable with this value.
  
• Simplify the resulting expression step-by-step.
  

In the given problem, we need to evaluate the rational function at

Calculating the Numerator

To find the value of the numerator:

• square
  
  
  
  
• Then, subtract 5 from this result.
  
  
  
  
  
  
  
  
  
  
  
  
  
  

  Calculating the Denominator

  
  To find the denominator value:
  
  
  
  
  
  
  
  Finally, place the simplified numerator and denominator back into the rational function.
  Check that
  
  
  
  
  
  

simplifying expressions

Simplifying expressions is a fundamental skill in algebra. It involves reducing an expression to its simplest form. Here are the general steps for simplifying:

• Combine like terms.
  
  
  
• Apply arithmetic operations carefully.
  
  
  

In the problem, after substituting
y = -2: The simplified numerator is






checking if the rational function is valid: since the resulting denominator is not zero It is valid to proceed with the division, resulting in

reducing a fraction.