Problem 19 - Elementary and Intermediate Algebra

Question

Evaluate each expression for \(y=-3 .\) See Example 1. $$ -\frac{y}{y^{2}-y+6} $$

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{1}{6} \).

Step 1: Substitute the Value of y

Start by substituting the given value of \( y = -3 \) into the expression.The expression is:\[ -\frac{y}{y^2 - y + 6} \]Substitute \( y = -3 \):\[ -\frac{-3}{(-3)^2 - (-3) + 6} \]

Step 2: Simplify the Denominator

Calculate the denominator by substituting \( y = -3 \):Calculate \((-3)^2\):\[ (-3)^2 = 9 \]Now, substitute into the expression for the denominator:\[ 9 - (-3) + 6 \]Simplify inside the brackets:\[ 9 + 3 + 6 = 18 \]

Step 3: Simplify the Expression

Now that we have the denominator as 18, substitute back into the expression:\[ -\frac{-3}{18} \]The negative sign at the front makes the fraction positive:\[ \frac{3}{18} \]

Step 4: Simplify the Fraction

Finally, simplify the fraction:Both 3 and 18 are divisible by 3:\[ \frac{3}{18} = \frac{1}{6} \]

Key Concepts

Substitution Method

The substitution method is a fundamental technique used to evaluate algebraic expressions by replacing variables with specific numeric values. This method helps to simplify expressions which might otherwise seem complex.

Substitution involves the following steps:

• Identify the variable in the expression and note its given value.
• Replace each occurrence of the variable in the expression with its given value.
• Proceed to evaluate or simplify the expression as needed.

By substituting the variable with a number, you essentially eliminate the unknown from the expression, making it much easier to handle. This technique is commonly used in solving equations where you need to find out if certain values satisfy an equation.

For example, in the original problem, we used the substitution method to evaluate the expression \(-\frac{y}{y^2 - y + 6}\) with \(y = -3\). This replacement simplified the problem to straightforward numerical calculations.

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest, most understandable form. This is done by finding common factors of the numerator and the denominator and dividing them by these shared factors.

The main steps in fraction simplification are:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.
• The resulting fraction should be equivalent to the original, just simpler.

For example, when simplifying \(\frac{3}{18}\), we note that both 3 and 18 can be divided by 3. Dividing both gives us \(\frac{1}{6}\), which is the simplest form of the fraction.

Simplifying fractions is crucial because it allows for easier interpretation and calculation, reducing the possibility of errors in subsequent mathematical operations. So, always look for ways to make fractions as simple as possible!

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operation symbols that represent a specific value or set of values. Understanding these expressions is key to solving equations and performing calculations accurately.

Important aspects of mathematical expressions include:

• The proper use of operations such as addition, subtraction, multiplication, and division.
• The need to follow the order of operations often remembered as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• Recognizing variables and understanding their role as placeholders for numbers.

Expressions can range from simple, like \(2 + 3\), to very complex, involving multiple operations and variables such as \(-\frac{y}{y^2 - y + 6}\). Regardless of complexity, the core principles of operation and order still apply.

When handling expressions, maintain clarity by simplifying wherever possible and using arithmetic rules judiciously to ensure accurate results.