Problem 49 - Prealgebra

Question

Find the value of each of the following expressions when \(x = 5\). $$-4 x+1$$

Step-by-Step Solution

Answer

The value of the expression is \(-19\).

Step 1: Substitute the Value

Replace the variable \(x\) with the number 5 in the expression \(-4x + 1\). This gives: \(-4 \times 5 + 1\).

Step 2: Perform Multiplication

Calculate \(-4 \times 5\), which equals \(-20\). So, the expression becomes \(-20 + 1\).

Step 3: Perform Addition

Add \(-20\) and \(1\) together to get \(-19\). Thus, the value of the expression is \(-19\).

Key Concepts

Substitution

Substitution is a fundamental concept in algebra that involves replacing variables in an expression with their corresponding numerical values. This technique enables us to simplify and evaluate expressions to find their numerical value. In the context of our exercise, substitution plays a crucial role. The expression we're working with is \(-4x + 1\). Before we can do anything else, we must substitute the given value of the variable, which is \(x = 5\), into the expression.To perform substitution:

• Identify the variable in the expression. Here, it is \(x\).
• Replace each occurrence of that variable with the given number. Thus, replace \(x\) with 5, transforming the expression from \(-4x + 1\) to \(-4 \times 5 + 1\).

This process is straightforward but essential, setting the stage for effective evaluation of the expression. Substitution simplifies the problem to a purely numerical one, allowing us to use arithmetic operations to find the final answer.

Order of Operations

The order of operations is a critical rule in mathematics that guides us on the correct sequence to solve parts of a mathematical expression. Remembering the order is simple with the help of the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Following this order ensures that everyone evaluates mathematical expressions consistently and correctly.In our expression \(-4 \times 5 + 1\), the order of operations dictates that multiplication should be addressed before addition:

• Multiplication: First, calculate \(-4 \times 5\). According to PEMDAS, do this before moving to any other operations. The result here is \(-20\).
• Addition: Next, look at any remaining operations, which is addition in this case. Add \(-20\) and \(1\) together to get the final result of \(-19\).

By sticking to the order of operations, we maintain accuracy in calculating expressions.

Integer Arithmetic

Integer arithmetic involves performing arithmetic operations using whole numbers, which can be positive, negative, or zero. When calculating with integers, considering the rules of arithmetic for positive and negative numbers is essential to avoid errors.In our example, we encounter multiplication and addition using integers:

• Multiplying Integers: Here, we multiply \(-4\) and \(5\). The rule for multiplication is that a negative and a positive number always result in a negative number: thus, \(-4 \times 5 = -20\).
• Adding Integers: The operation \(-20 + 1\) requires adding a negative and a positive number. When adding such numbers, think of it as combining their absolute values. Subtract the smaller absolute value from the larger one and keep the sign of the number with the larger absolute value, resulting in \(-19\).

Mastering these basic integer rules is vital for evaluating expressions accurately and efficiently. It provides a solid foundation for more advanced mathematical concepts.