Problem 18 - Elementary and Intermediate Algebra

Question

Evaluate each expression for \(y=-3 .\) See Example 1. $$ \frac{2 y+9}{y^{2}+25} $$

Step-by-Step Solution

Answer

The evaluated expression is \( \frac{3}{34} \).

Step 1: Substitute the Given Value

Substitute the given value, \( y = -3 \), into the expression \( \frac{2y + 9}{y^2 + 25} \). This gives us: \( \frac{2(-3) + 9}{(-3)^2 + 25} \).

Step 2: Simplify the Numerator

Calculate the value of the numerator by simplifying \( 2(-3) + 9 \). This results in \( -6 + 9 = 3 \).

Step 3: Simplify the Denominator

Calculate the value of the denominator by simplifying \( (-3)^2 + 25 \). This results in \( 9 + 25 = 34 \).

Step 4: Evaluate the Expression

Combine the simplified values from the previous steps to evaluate the expression: \( \frac{3}{34} \). This is the final value of the expression.

Key Concepts

Substitution Method

The substitution method is a powerful tool in algebra that allows you to evaluate an expression by plugging in given values for the variables. When you're tasked with evaluating an expression, the first step is to identify which variable needs to be substituted.

• First, locate the variable in the expression.
• Next, replace the variable with the given value.

For example, in the expression \( \frac{2y + 9}{y^2 + 25} \), if you're given \( y = -3 \), you plug \( -3 \) into the places where \( y \) appears. This simplifies your work by turning a variable equation into one only involving numbers, making it easier to solve. The calculations then become a matter of standard arithmetic operations.

Simplifying Expressions

Simplifying expressions means reducing an expression to its simplest form. This includes performing all possible arithmetic operations and reducing fractions if necessary. When simplifying, focus on one part of the expression at a time:

• Begin with the numerator or the terms on the top.
• Calculate the result by following arithmetic operations: addition, subtraction, multiplication, or division.
• Do the same for the denominator or the terms at the bottom.

In our example, to simplify the expression \( \frac{2(-3) + 9}{(-3)^2 + 25} \), we first simplify the numerator \( 2(-3) + 9 = -6 + 9 = 3 \). Then, simplify the denominator \( (-3)^2 + 25 = 9 + 25 = 34 \). The final step is understanding that the fraction \( \frac{3}{34} \) is already in its simplest form since 3 and 34 share no common factors.

Algebraic Fractions

Algebraic fractions are just fractions that contain variables in the numerator, the denominator, or both. They behave similarly to regular fractions but often need special handling due to the presence of variables. Key aspects to consider include:

• Ensure the fraction is expressed in its simplest form.
• Factor if needed, although not required in this example.
• Verify calculations carefully when dealing with variables.

In the example \( \frac{2y + 9}{y^2 + 25} \), understanding that substituting and then simplifying was enough to arrive at \( \frac{3}{34} \) is crucial. Although this is a numeric fraction now, remembering it came from an algebraic one is important, so similar techniques apply to more complex or variable-intense problems. Always double-check whether further simplification or factorizing might be applicable.