Problem 53 - Prealgebra

Question

Evaluate each expression. $$b+11, b=-15$$

Step-by-Step Solution

Answer

The value of the expression is -4.

Step 1: Substitute the Given Value

Replace the variable \( b \) in the expression \( b + 11 \) with the given value of \( b = -15 \). This changes the expression to \( -15 + 11 \).

Step 2: Simplify the Expression

Perform the arithmetic operation by adding \(-15\) and \(11\). This is done by understanding that adding a positive number to a negative number involves reducing the negative number. So, \(-15 + 11 = -4\).

Key Concepts

Variable Substitution

In arithmetic and algebra, the process of variable substitution refers to replacing variables in an expression or equation with their given values. This is a fundamental step when you need to evaluate expressions because it translates a variable-based equation into one that only involves numbers. Let's use our example to illustrate this process.

Given the expression \(b + 11\), where \(b = -15\), we need to substitute \(b\) with \(-15\). This turns the expression from using a letter to using numbers, changing it to \(-15 + 11\).

Variable substitution is crucial because it simplifies expressions and prepares them for further arithmetic operations.

Integer Operations

Working with integer operations involves adding, subtracting, multiplying, and dividing numbers without fractions or decimals. In our exercise, we performed integer addition, which combines both positive and negative numbers.

Let's break down the operation \(-15 + 11\). Here, you have:

• A negative integer (\(-15\))
• A positive integer (\(11\))

When adding integers with different signs, you subtract the smaller absolute value from the larger absolute value and give the result the sign of the number with the larger absolute value. Here’s how it works:

• Find the absolute values: |\(-15\)| = 15 and |\(11\)| = 11
• Subtract 11 from 15 to get 4
• Since the larger absolute value was initially negative, the result is \(-4\)

Expression Evaluation

Expression evaluation is the final step where we calculate a specific value for an expression using arithmetic operations and the values substituted for variables. This means following standard arithmetic rules to solve completely.

In the explored exercise, we evaluated \(-15 + 11\) through integer operations. Since expressions involve combining numbers and operations, understanding order and procedure is key. After substitution and arithmetic operations, we found the value of the expression to be \(-4\).

Expression evaluation is an essential skill as it allows us to derive numerical answers from algebraic expressions, turning theoretical problems into solved situations.