Problem 93 - Elementary Algebra

Question

For Problems \(93-106\), evaluate each algebraic expression for the given values of the variables. $$ 4 x^{2}+7 y^{2} \text { for } x=-2 \text { and } y=-3 $$

Step-by-Step Solution

Answer

The expression evaluates to 79.

Step 1: Substitute given values

First, substitute the given values of the variables \( x = -2 \) and \( y = -3 \) into the algebraic expression \( 4x^2 + 7y^2 \). This gives us \( 4(-2)^2 + 7(-3)^2 \).

Step 2: Calculate the squares

Calculate \((-2)^2\) and \((-3)^2\). We have:1. \((-2)^2 = 4\)2. \((-3)^2 = 9\)

Step 3: Multiply by coefficients

Multiply the results of the squares by the coefficients in the expression:1. \(4 \times 4 = 16\)2. \(7 \times 9 = 63\)

Step 4: Add the results

Add the results of the multiplied values: \(16 + 63 = 79\).

Key Concepts

Substitution Method in Algebra

The substitution method in algebra is a technique used to simplify algebraic expressions and equations by replacing variables with given numerical values. This method is crucial when we need to find the value of an expression for specific variable values. By substituting these values into the expression, we transform it into a numerical form that is easier to manage.
When dealing with substitution:

• Identify the variables in the expression.
• Replace each variable with its assigned numerical value.
• Follow the order of operations (PEMDAS/BODMAS) to correctly solve the numeric expression.

For example, with the expression \(4x^2 + 7y^2\), and given \(x = -2\) and \(y = -3\), substitution means replacing \(x\) with \(-2\) and \(y\) with \(-3\). Thus, the expression becomes \(4(-2)^2 + 7(-3)^2\).
This method is foundational for evaluating algebraic expressions accurately and lays the groundwork for solving more complex algebraic equations.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators (such as +, -, ×, ÷) that represent some quantity. They are the building blocks of algebra and provide a structured way to solve problems mathematically.
Expressions can vary in complexity:

• Simple Expressions: Involve basic arithmetic, like \(3 + 5\) or \(7x\).
• Polynomial Expressions: Contain multiple terms, often with variables raised to different powers, e.g., \(4x^2 + 7y^2\).
• Rational Expressions: Represent fractions involving variables, like \(\frac{x+1}{y}\).

Understanding expressions means knowing how to combine like terms, factor, and simplify. In the provided problem, \(4x^2 + 7y^2\) is a polynomial expression. Evaluating such expressions requires careful manipulation of terms and adherence to the rules of arithmetic and algebra. The goal is to simplify the expression into a single, computable value.

Algebraic Squares

Algebraic squares refer to an algebraic expression raised to the power of two, symbolized as \(x^2\) or \(y^2\). Recognizing and calculating squares is essential because they appear frequently in algebra.
When you square a number or variable:

• Multiply the number by itself: the square of \(x\) is \(x \times x\), represented as \(x^2\).
• For negative numbers, squaring results in a positive value because negative \(\times\) negative = positive (e.g., \((-2)^2 = 4\)).

In the exercise, we calculated \((-2)^2\) and \((-3)^2\) during the substitution process, obtaining 4 and 9, respectively. This allows us to transform the algebraic expression into a more workable numeric form.
Thus, having a strong grasp of squaring techniques speeds up the process of evaluating complex algebraic expressions.