Problem 87 - Elementary and Intermediate Algebra

Question

Evaluate each expression, for \(x=3, y=-2,\) and \(z=-4\) See Example 10. $$ -\frac{2 x+y^{3}}{y+2 z} $$

Step-by-Step Solution

Answer

The evaluated expression is \(\frac{1}{5}\).

Step 1: Substitute Values

Given the expression \(-\frac{2x + y^3}{y + 2z}\), first substitute \(x = 3\), \(y = -2\), and \(z = -4\) into the expression. This gives us: \[-\frac{2(3) + (-2)^3}{-2 + 2(-4)}\].

Step 2: Evaluate the Numerator

Calculate the numerator by computing each part: 1. Calculate \(2(3)\), which is \(6\).2. Calculate \((-2)^3\), which is \(-8\).3. Add these results: \(6 + (-8) = -2\). So, the numerator is \(-2\).

Step 3: Evaluate the Denominator

Calculate the denominator by computing each part: 1. Calculate \(2(-4)\), which is \(-8\).2. Add \(-2 + (-8) = -10\). So, the denominator is \(-10\).

Step 4: Simplify the Expression

Substitute the calculated numerator and denominator back into the expression: \[-\frac{-2}{-10}\].We can simplify this by recognizing that a negative divided by a negative is positive: \[\frac{2}{10} = \frac{1}{5}\].

Step 5: Final Answer

Thus, the evaluated expression \(-\frac{2x + y^3}{y + 2z}\) for \(x = 3\), \(y = -2\), and \(z = -4\) is \(\frac{1}{5}\).

Key Concepts

Substitution Method

The substitution method is a fundamental technique in algebra. It involves replacing variables in an expression with their given values to simplify and resolve the equation. This is particularly handy when you are given specific values for the variables, like in the exercise.
Here's how you can apply this:

• Identify the variables in your expression. In our original expression, these are \(x\), \(y\), and \(z\).
• Replace each variable with its corresponding given value. For example, substitute \(x = 3\), \(y = -2\), and \(z = -4\) into the expression \(-\frac{2x + y^3}{y + 2z}\).
• Rewrite the expression with these values. It becomes: \(-\frac{2(3) + (-2)^3}{-2 + 2(-4)}\).

This substitution transforms the original abstract expression into a numerical one that you can evaluate step by step.

Numerator and Denominator

Understanding the terms "numerator" and "denominator" is key to working with fractions. In any fraction, the numerator is the top part, and the denominator is the bottom part. For the fraction \(-\frac{2x + y^3}{y + 2z}\), knowing how to evaluate each part is crucial.

Numerator

This is the part of the fraction above the division line:

• First, calculate \(2x\) with \(x = 3\), so \(2(3) = 6\).
• Next, compute \(y^3\) with \(y = -2\), thus \((-2)^3 = -8\).
• Finally, sum these results: 6 + (-8) = -2.

So, the numerator becomes \(-2\).

Denominator

This is the part under the division line:

• Calculate \(2z\) with \(z = -4\), which is \(2(-4) = -8\).
• Sum these with \(y = -2\), resulting in \(-2 + (-8) = -10\).

Thus, the denominator is \(-10\). Understanding these concepts helps in transitioning from an algebraic expression to a numerical fraction.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form, where the numerator and the denominator share no common factors other than 1. In this exercise, after finding the fraction \(-\frac{-2}{-10}\), it's important to simplify it correctly.

Here’s how that process works:

• Recognize that dividing a negative by another negative yields a positive result: \(-\frac{-2}{-10}\) becomes \(\frac{2}{10}\).
• Find the greatest common divisor (GCD) of the numerator and the denominator. Here, the GCD of 2 and 10 is 2.
• Divide both the numerator and the denominator by their GCD: \(\frac{2 \div 2}{10 \div 2} = \frac{1}{5}\).

This process results in the simplest form of the fraction, \(\frac{1}{5}\). Learning to simplify fractions ensures that you can present your answers in their most straightforward form, which is often required in both mathematics classes and exams.