Problem 65 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The value of the expression is 21.

Step 1: Substitute Given Values

First, we substitute the given values into the expression. The values given are \(x = -5\), \(y = 4\), and \(t = 10\). In the expression \(y^2 - x\), substitute \(y = 4\) and \(x = -5\). This gives us \((4)^2 - (-5)\).

Step 2: Calculate the Square of y

Calculate \(4^2\). When we square 4, we get \(4 \times 4 = 16\). So substituting back, the expression becomes \(16 - (-5)\).

Step 3: Simplify the Expression

Subtract \(-5\) from 16. Subtraction of a negative number is the same as addition, therefore \(16 - (-5) = 16 + 5 = 21\).

Key Concepts

Substitution Method

The substitution method is a crucial concept in algebra that allows us to solve expressions or equations by replacing variables with given numerical values. It is the first step when working with any algebraic expression, and it's essential to carry it out accurately to avoid errors in the later stages of calculation.
For example, to evaluate the expression \(y^2 - x\) given \(x = -5\) and \(y = 4\), we substitute these values directly into the expression. This converts \(y^2 - x\) into \((4)^2 - (-5)\).
It's like if each variable is a placeholder. When we perform substitution, we fill in these placeholders with the actual numbers provided in the problem.

Squaring Numbers

Squaring a number is a specific arithmetic operation where you multiply a number by itself. It's a fundamental skill you need when working with expressions that involve exponents, such as \(y^2\).
In our example, \(y = 4\), so \(y^2\) means evaluating \(4^2\). Calculating \(4^2\) involves multiplying 4 by itself, leading to \(4 \times 4 = 16\).
Squaring numbers is not only limited to integers; it can apply to any number or algebraic expression. Understanding squaring deeply ensures you can deal efficiently with more complex expressions or problems.

Simplifying Expressions

Simplifying expressions is about breaking them down into their simplest form for easier understanding and interpretation. In algebra, this often involves performing operations such as addition, subtraction, multiplication, or division to streamline the expression.
After substituting and squaring in our example, we're left with \(16 - (-5)\). A critical part of simplifying here is understanding how subtraction and negative numbers interact, allowing us to rewrite \(16 - (-5)\) as \(16 + 5\) which simplifies to 21.
Simplifying helps not only in solving expressions but also in verifying correctness and making computations easier.

Negative Numbers

Negative numbers can seem tricky but are straightforward once you understand the basic rules that govern their operations. In the exercise, negative numbers appear in both substitution and simplification steps.
One such rule is subtracting a negative number, which is equivalent to adding its positive counterpart. Thus, in the case of \(16 - (-5)\), this operation is transformed into \(16 + 5\).
These characteristics of negative numbers are crucial for correctly handling operations in algebraic expressions, ensuring accurate computations in exercises like the one provided. Understanding them gives you confidence in manipulating equations that might initially seem complex.