Problem 59 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The value is \(-9\).

Step 1: Substitute Values

First, substitute the given values of the variables into the expression. Here, replace \(x\) with \(-5\) and \(y\) with \(4\). This transforms the expression \(x - y\) into \(-5 - 4\).

Step 2: Perform the Subtraction

Calculate the value of the expression \(-5 - 4\). Since both numbers are negative, add their absolute values, which gives you \(5 + 4 = 9\), and keep the negative sign. Thus, the result is \(-9\).

Key Concepts

Substitution Method

The substitution method in algebra involves replacing variables with their given numerical values. This technique is crucial when we need to evaluate expressions with specific values assigned to variables. For instance, if we have an expression involving variables like \(x\) and \(y\), and we are given specific values for these variables, our first step is to substitute these values into the expression.
To apply the substitution:

• Identify the variables and their corresponding values.
• Replace each variable in the expression with its value.
• Once substitution is complete, proceed to simplify or evaluate the expression.

In our example, the expression \(x-y\) involves variables \(x\) and \(y\). By substituting \(x = -5\) and \(y = 4\) into the expression, it transforms to \(-5 - 4\). The substitution method is essential for evaluating expressions with precision, ensuring that the variable values are correctly plugged into the algebraic expression.

Evaluating Expressions

Evaluating expressions involves simplifying them by performing the required mathematical operations after substituting values for the variables. Once you've completed the substitution step, it's time to execute the operations involving the substituted values to find the expression's value. To evaluate an expression:

• Follow the order of operations: parentheses, exponents, multiplication and division (from left to right), addition and subtraction (from left to right), often remembered by the acronym PEMDAS.
• After the substitution step, simply carry out these operations in the correct order.

In the example \(-5 - 4\), no additional operations or parenthetical elements are involved, so simply perform the subtraction. Negative and positive values are managed by the rules of integer operations, leading to the final value \(-9\). Evaluating expressions effectively requires careful attention to order and accuracy in processing mathematical operations.

Integer Operations

Integer operations are fundamental when dealing with algebraic expressions, especially those involving addition and subtraction. Integer operations ensure we accurately handle positive and negative numbers during calculations. The basics of integer operations include:

• When subtracting integers, consider the subtraction as adding the opposite. For instance, \(a - b\) can be thought of as \(a + (-b)\).
• If both integers are negative, as in our given example \(-5 - 4 \), add their absolute values and keep the negative sign. Thus, it becomes \(5 + 4 = 9\), and we retain the negative sign, leading to \(-9\).

Understanding integer operations simplifies problem-solving processes and provides accuracy in evaluating expressions like those in algebra, ensuring confidence in your solutions.