Problem 65 - Beginning and Intermediate Algebra

Question

Evaluate each expression when \(x=-5, y=4,\) and \(t=10 .\) See Example 6. $$ y^{2}-x $$

Step-by-Step Solution

Answer

The value of the expression is 21.

Step 1: Identify Variables

Firstly, let's identify the given values for the variables. We have \( x = -5 \), \( y = 4 \), and \( t = 10 \). However, since \( t \) is not used in the expression, it can be ignored for this evaluation.

Step 2: Substitute Values

Substitute \( x = -5 \) and \( y = 4 \) in the expression \( y^{2} - x \). This gives us: \( 4^{2} - (-5) \).

Step 3: Evaluate Squaring Term

Calculate \( 4^{2} \). Squaring 4 gives 16. Thus, the expression becomes \( 16 - (-5) \).

Step 4: Evaluate the Expression

The expression \( 16 - (-5) \) can be simplified by recognizing that subtracting a negative number is the same as adding its positive. Therefore, it becomes \( 16 + 5 \). This results in 21.

Key Concepts

Variable Substitution

Variable substitution is an important initial step when evaluating algebraic expressions. It involves replacing variables within an expression with their given numeric values. For example, in the exercise given, the expression to evaluate is \( y^2 - x \). We know that \( x = -5 \) and \( y = 4 \). By substituting these values into the expression, we get \( 4^2 - (-5) \). This step is crucial because it transforms an algebraic expression into a numerical one, allowing us to use arithmetic operations to simplify and solve it. Without correct substitution, the evaluation of the expression will not yield a valid result. This part of the process sets the stage for the next steps, which involve computation and simplification of the expression.

Squaring Numbers

Squaring a number is a mathematical operation where a number is multiplied by itself. It's a common and simple operation often encountered when dealing with quadratic expressions or evaluating specific power terms.In this case, the value of \( y \) is 4, and its square is denoted by \( y^2 \). Calculating \( 4^2 \) involves multiplying 4 by 4, which results in 16. Understanding squaring is crucial because it directly impacts the value of an expression. If squaring is not done correctly, it may lead to errors in final calculations. Squares of numbers are always non-negative, and they grow rapidly as numbers become larger.✅ Here are a few fun facts about squaring:

• The square of a negative number is always positive because multiplying two negative numbers results in a positive number.
• Zero squared is still zero, because zero times any number equals zero.

Knowledge of this operation helps simplify and compute expressions accurately.

Evaluating Expressions

Once variable substitution and squaring have been completed, evaluating the expression involves performing all arithmetic operations in the correct order.In our expression \( 16 - (-5) \), we recognize that subtracting a negative number is equivalent to adding its positive counterpart. Therefore, this expression simplifies to \( 16 + 5 \), which equals 21.When evaluating expressions:

• Follow the order of operations, PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• Be mindful of positive and negative signs, as they greatly influence the resulting value.
• Double-check calculations for accuracy.

Evaluation allows us to simplify expressions and find their numerical values. This final step tests your understanding of arithmetic operations and sign rules, confirming your computation was executed properly.