Problem 96 - Elementary Algebra

Question

For Problems \(93-106\), evaluate each algebraic expression for the given values of the variables. $$ x^{2}-2 y^{2} \text { for } x=-\frac{2}{3} \text { and } y=\frac{3}{2} $$

Step-by-Step Solution

Answer

The expression evaluates to \(-\frac{73}{18}\).

Step 1: Substitute the given values into the expression

We have the algebraic expression \(x^{2} - 2y^{2}\), and we are given \(x = -\frac{2}{3}\) and \(y = \frac{3}{2}\). We substitute these values into the expression:\[x^{2} - 2y^{2} = \left(-\frac{2}{3}\right)^{2} - 2\left(\frac{3}{2}\right)^{2}\]

Step 2: Evaluate the squares of the variables

Compute the square of each value. First, calculate \((-\frac{2}{3})^{2}\), which is:\[\left(-\frac{2}{3}\right)^{2} = \frac{4}{9}\]Then, calculate \((\frac{3}{2})^{2}\), which is:\[\left(\frac{3}{2}\right)^{2} = \frac{9}{4}\]

Step 3: Multiply the second term by 2

Recall the expression includes \(-2y^{2}\). Substitute \(y^{2}\) with \(\frac{9}{4}\) and multiply by 2:\[2 \times \frac{9}{4} = \frac{18}{4} = \frac{9}{2}\]

Step 4: Perform the subtraction

Subtract \(\frac{9}{2}\) from \(\frac{4}{9}\).First, express \(\frac{9}{2}\) with a common denominator \(18\) for the subtraction:\[\frac{4}{9} = \frac{8}{18} \text{ and } \frac{9}{2} = \frac{81}{18}\]Now, perform the subtraction:\[\frac{8}{18} - \frac{81}{18} = -\frac{73}{18}\]

Step 5: Simplify the result (if possible)

The result \(-\frac{73}{18}\) is already in its simplest form since 73 is a prime number and does not share any common factors with 18 except 1.

Key Concepts

Substitution in Algebra

Substitution in algebra involves replacing variables in an algebraic expression with their given numerical values. To understand this better, consider the expression given in the exercise:

• Expression: \(x^2 - 2y^2\)
• Given Values: \(x = -\frac{2}{3} \) and \(y = \frac{3}{2} \)

Here, substituting means you will replace \(x\) with \(-\frac{2}{3}\) and \(y\) with \(\frac{3}{2}\) in the entire expression.
The act of substitution transforms the original algebraic expression into a numeric form: \[(-\frac{2}{3})^2 - 2(\frac{3}{2})^2\]This step lays the foundation for evaluating the expression into a single number. It's important because it converts abstract expressions into computations that can be systematically worked out.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q eq 0\). In our exercise, the values to substitute for \(x\) and \(y\) are both rational numbers:

• The value \(x = -\frac{2}{3}\) is a rational number because it is expressed as a fraction.
• Similarly, \(y = \frac{3}{2}\) is also a rational number.

Using rational numbers involves arithmetic operations such as addition, subtraction, multiplication, and division with fractions. Calculating results requires common denominators for addition and subtraction, as well as a solid understanding of fraction multiplication and simplification processes. This exercise specifically demonstrates multiplication of rational numbers when calculating powers and performing multiplications:

• Squaring \((-\frac{2}{3})\) and \((\frac{3}{2})\).
• Multiplying \(2 \cdot (\frac{9}{4})\).
• Finding a common denominator to further simplify results post subtraction.

Order of Operations

The order of operations is a set of rules that determines the correct sequence to evaluate a mathematical expression. Known as PEMDAS/BODMAS, it stands for:

• P/B: Parentheses/Brackets
• E/O: Exponents/Orders (such as squares and square roots)
• M/D: Multiplication and Division (from left to right)
• A/S: Addition and Subtraction (from left to right)

Applying these rules in the exercise ensures the terms are dealt with correctly:
1. **Calculate the Exponents**: Start by squaring the substituted values, such as \((-\frac{2}{3})^2 = \frac{4}{9}\) and \((\frac{3}{2})^2 = \frac{9}{4}\).
2. **Complete Multiplication**: You then multiply the result of \(y^2\) by \(-2\). This follows because of the multiplication aspect of the \(-2y^2\) part in the expression.
3. **Perform Subtraction**: Finally, you subtract \(\frac{9}{2}\) from \(\frac{4}{9}\), ensuring to find a common denominator before performing this step, leading to \(\frac{8}{18} - \frac{81}{18}\).
Without following the order of operations, the calculation might lead to incorrect results by misplacing operations or evaluating in the wrong sequence.