Problem 12 - Prealgebra

Question

Evaluate each expression if \(x=-2\) and \(y=4\) $$x^{2}+y^{2}$$

Step-by-Step Solution

Answer

The expression evaluates to 20.

Step 1: Substitute the values of x and y

Substitute the given values of \(x = -2\) and \(y = 4\) into the expression \(x^2 + y^2\). This gives us \((-2)^2 + (4)^2\).

Step 2: Calculate the squares

Calculate the square of each term independently: \((-2)^2\) is \(4\), and \((4)^2\) is \(16\).

Step 3: Add the results

Add the results from the previous step: \(4 + 16 = 20\).

Key Concepts

Expressions Evaluation

In prealgebra, evaluating expressions is a fundamental skill. It involves replacing variables in an expression with given numbers and performing the necessary calculations to find the answer. This process helps us analyze and simplify problems to find specific numerical results. In our example, we evaluated the expression \(x^2 + y^2\) when \(x = -2\) and \(y = 4\). Evaluating expressions requires focusing on:

• Identifying the variables in the expression.
• Knowing the values assigned to these variables.
• Substituting the variables with their given values.
• Performing mathematical operations like addition or multiplication.

Remember, following a consistent order of operations is crucial to get the correct result. Expressions evaluation is a powerful tool for solving a range of math problems efficiently.

Substitution in Expressions

Substitution is the process of replacing variables in an algebraic expression with specific numbers. This method transforms an abstract expression into a concrete numerical problem, making it easier to solve. Let's take a closer look at how this works:

• Identify Variables: Begin by determining which letters (variables) you need to substitute. In our exercise, these are \(x\) and \(y\).
• Insert Values: Replace each variable with the provided number. Using our example, substitute \(x = -2\) and \(y = 4\) into \(x^2 + y^2\), which will become \((-2)^2 + 4^2\).
• Rewrite Expression: Once substituted, rewrite the expression with the numbers in place of variables for clarity.

Substitution might seem straightforward, but it requires attention to detail to ensure nothing is missed. This skill is foundational in solving algebraic expressions and can simplify complex problems into manageable computations.

Square Calculation

Calculating squares of numbers is a basic yet essential math skill, especially in algebra. A square of a number is simply that number multiplied by itself. Here's how you perform square calculations:

• Understand the Concept: Squaring a number, written as \(n^2\), means multiplying the number by itself. So, \((-2)^2\) means multiplying \(-2\) by \(-2\), which is \(4\).
• Positive Results: When you square negative numbers, the result is always positive because a negative times a negative equals a positive.
• Practice with Positive Numbers: For any positive number like \(4\), \(4^2\) equals \(16\) because \(4 \times 4 = 16\).

Mastering square calculations helps in solving a variety of problems, from simple arithmetic to more complex algebraic equations. It's crucial for students to become comfortable with squares to prepare for more advanced math concepts.