Problem 60 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The value of the expression is 9.

Step 1: Identify the Variables

First, identify the given variables from the problem. We have the values: \( x = -5 \), \( y = 4 \), and \( t = 10 \). For this expression, we will only use \( x \) and \( y \).

Step 2: Substitute the Values

Substitute the values of the variables into the expression \( y - x \). This means replacing \( y \) with 4 and \( x \) with -5.

Step 3: Perform the Calculation

Now substitute these values into the expression: \( y - x = 4 - (-5) \). Since subtracting a negative is equivalent to adding, the expression becomes \( 4 + 5 \).

Step 4: Simplify the Expression

Simplify the expression \( 4 + 5 \) to get the final result. The result of this calculation is 9.

Key Concepts

Variable Substitution

In algebra, variable substitution is a powerful technique that allows us to replace variables with their actual values. This makes abstract algebraic expressions easier to evaluate. For students tackling a problem like evaluating the expression \( y - x \), the first step is identifying the values of the variables involved.

In this scenario, we were provided with the values \( x = -5 \) and \( y = 4 \). Our task is to substitute these values into the expression. This involves replacing \( y \) with 4 and \( x \) with -5 in the expression \( y - x \).

Here's how you do it:

• The original expression is \( y - x \).
• Replace \( y \) with 4, resulting in \( 4 - x \).
• Next, replace \( x \) with -5. This changes the expression to \( 4 - (-5) \).

It's important to handle negatives carefully during substitution, as they can change the operation from subtraction to addition.

Integer Arithmetic

Once the variables are successfully substituted, we perform integer arithmetic to calculate the expression. Integer arithmetic involves basic operations such as addition, subtraction, multiplication, and division of whole numbers.

For the expression \( 4 - (-5) \), recognize that subtracting a negative number is the same as adding its absolute value. In simple terms, \( 4 - (-5) \) transforms into \( 4 + 5 \). This particular rule is handy and frequently used in algebraic evaluations.

Breaking it down:

• We start with \( 4 - (-5) \).
• Changing the negative subtraction to positive addition gives us \( 4 + 5 \).

These steps might appear basic, but mastering them ensures a smooth handling of more complex mathematical problems in the future.

Expression Simplification

The final step in evaluating an algebraic expression involves expression simplification. This step aims to reduce the expression to its simplest form, or a single value, for easier interpretation and insight.

After substituting variables and transforming the operations in the expression \( 4 + 5 \), the task is to perform the addition. Calculating \( 4 + 5 \) gives us 9. Through the process of simplification, the expression \( y - x \) is evaluated to a single integer value.

Key steps in expression simplification typically include:

• Carrying out arithmetic operations sequentially.
• Reducing mathematical expressions to their simplest numerical form, if possible.

The simplicity of this operation demonstrates the essence of expression simplification: making a complex problem more manageable and understandable.