Problem 133 - Elementary and Intermediate Algebra

Question

Evaluate each expression for \(x=-3\) and \(y=-5\) $$ \frac{x-y^{2}}{2 y-1+x} $$

Step-by-Step Solution

Answer

The value of the expression is 2.

Step 1: Substitute Values

First, substitute the given values of \(x = -3\) and \(y = -5\) into the expression \(\frac{x-y^{2}}{2y-1+x}\). This will give us \(\frac{-3 - (-5)^{2}}{2(-5) - 1 - 3}\).

Step 2: Calculate Numerator

In the numerator, calculate \(-3 - (-5)^2\). The value of \((-5)^2\) is 25, so the expression becomes \(-3 - 25 = -28\).

Step 3: Calculate Denominator

In the denominator, calculate \(2(-5) - 1 - 3\). This becomes \(-10 - 1 - 3 = -14\).

Step 4: Simplify Fraction

The expression is now \(\frac{-28}{-14}\). Simplify the fraction by noticing that both the numerator and the denominator are negative, resulting in \(\frac{28}{14}\). Divide 28 by 14 to get 2.

Key Concepts

Expression Evaluation

Evaluating an expression involves replacing variables with their given values to simplify and find the numerical result. In this exercise, we're working with the expression \(\frac{x-y^{2}}{2y-1+x}\) and need to evaluate it using \(x = -3\) and \(y = -5\). Here's a simple strategy:

• First, substitute the values into the expression, making it easier to work with pure numbers.
• Perform any operations like square calculations, additions, or subtractions following the order of operations (PEMDAS/BODMAS).
• After replacing the variables, simplify each part step-by-step to reach the solution.

As we can see, expression evaluation is a methodical process that helps in transforming abstract mathematical expressions into concrete numbers.

Substitution Method

The substitution method involves replacing variables in an expression or equation with their specific values, allowing us to find a numerical result. In our exercise, the substitution process began with replacing \(x = -3\) and \(y = -5\) directly into the given algebraic expression: \[\frac{x-y^{2}}{2y-1+x}\]This results in plugging these values into the expression, transforming it into numerical form: \[\frac{-3 - (-5)^{2}}{2(-5) - 1 - 3}\]Using substitution is essentially connecting the abstract algebraic representation to real numbers, making it easier to perform subsequent calculations. The key advantage here is clarity and ease in solving complex expressions.

• Always double-check the plugged-in values to avoid mistakes.
• Carefully follow mathematical operations to ensure accuracy.

Simplifying Fractions

Simplifying fractions is a practical skill in algebra that helps reduce fractions to their simplest form, making them easier to interpret. Once you've evaluated an expression's numerator and denominator, the next step is to simplify the resulting fraction. Consider this scenario as in the exercise solution:

• The expression led to the fraction \(\frac{-28}{-14}\).
• Noticing both the numerator and denominator are negative, we take advantage to make them positive: \(\frac{28}{14}\).
• Then, divide the numbers by their greatest common divisor (14 in this case) to simplify: 28 divided by 14 gives 2.

These steps highlight how simplifying fractions is crucial for reducing algebraic results to the simplest, comprehensible form. Mastering this process ensures clarity and efficiency in handling algebraic expressions.