Problem 66 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The value of the expression is 105.

Step 1: Substitute Variables

Start by substituting the given values of the variables into the expression. We have: \[t = 10, x = -5\]Therefore, substitute these values into the expression:\[t^{2} - x ightarrow 10^{2} - (-5)\]

Step 2: Simplify the Power

Calculate the power of 10. In mathematics, a number raised to the power of 2 is simply the number multiplied by itself. Therefore:\[10^{2} = 10 \times 10 = 100\]

Step 3: Simplify the Subtraction

Next, simplify the expression by subtracting. Pay attention to the double negative:\[100 - (-5)\]Utilizing the rule that subtracting a negative number is the same as adding its absolute value, we get:\[100 + 5\]

Step 4: Final Calculation

Complete the calculation: \[100 + 5 = 105\]Thus, the result of the expression \( t^2 - x \) is 105.

Key Concepts

Substitution Method

The substitution method involves replacing the variables in an algebraic expression with given values. It is a straightforward technique that allows us to simplify expressions and find numerical equivalents. For example, consider the expression \( t^2 - x \). If we know \( t = 10 \) and \( x = -5 \), we substitute these values into the expression. This gives us \( 10^2 - (-5) \). This step is crucial because it transforms an algebraic expression into an arithmetic one, enabling us to perform further calculations with concrete numbers.

Simplifying Powers

Simplifying powers is an essential part of dealing with algebraic expressions, particularly when substitution involves exponents. When we simplify powers, such as \( 10^2 \), we multiply the base by itself the number of times indicated by the exponent. In this case, \( 10^2 \) means \( 10 \times 10 \), which equals \( 100 \). Simplifying powers helps break down complex expressions to more manageable numbers, making subsequent operations like addition or subtraction easier to handle.

Double Negative

Understanding the concept of double negatives in mathematics is vital. A double negative occurs when we have a negative sign in front of another negative sign. In arithmetic operations, a double negative essentially turns into a positive. For example, when simplifying \( 100 - (-5) \), the double negative causes us to actually add. So it turns into \( 100 + 5 \). This conversion is based on the rule that subtracting a negative number is equivalent to adding its positive counterpart.

Evaluation of Expressions

Evaluating expressions combines all steps, from substitution to simplification, in order to find a numerical result. After substituting values and simplifying components like powers and double negatives, we determine the value of the whole expression. In the example \( t^2 - x \), we substituted \( t = 10 \) and \( x = -5 \), resulting in \( 10^2 - (-5) \) which simplified to \( 100 + 5 \). The final outcome, \( 105 \), illustrates the power of methodically evaluating expressions to obtain solutions in algebra.