Problem 55 - Prealgebra

Question

Find the value of each of the following expressions when \(a = -2\). $$-7 a+4$$

Step-by-Step Solution

Answer

The value of the expression when \(a = -2\) is 18.

Step 1: Substitute the given value

First, we'll replace every instance of the variable 'a' in the expression with the given value, which is -2. The expression is \(-7a + 4\). By substituting, we get:\[-7(-2) + 4\].

Step 2: Multiply the constants with 'a'

Next, we need to multiply -7 with -2. Multiplying two negative numbers results in a positive number. So, calculate:\[-7 \times (-2) = 14\].

Step 3: Add the results

Now, we have the expression: \[14 + 4\]. Next, perform the addition:\[14 + 4 = 18\].

Key Concepts

Substitution

Substitution is the process of replacing variables with their actual values in an algebraic expression. In this exercise, we substitute the variable \( a \) with the value \( -2 \) in the expression \(-7a + 4\). This step is crucial because it transforms the expression into one that only contains numbers, making it easier to solve. To perform substitution, identify the variable and replace it wherever it appears in the expression. For example:

• Original expression: \(-7a + 4\)
• Substituting \( a = -2 \): \(-7(-2) + 4\)

This step ensures that all the mathematical operations will be done with real numbers, simplifying the solving process.

Multiplication of Integers

Understanding the multiplication of integers, especially when dealing with negative numbers, is a fundamental concept in prealgebra. Multiplying two numbers involves combining them into a single product. When these numbers are both negative, such as \(-7\) and \(-2\), their product becomes positive. This is because:

• The product of two positive numbers is positive.
• The product of a positive and a negative number is negative.
• The product of two negative numbers is positive.

For the expression \(-7(-2) + 4\), we first calculate \(-7 \times (-2)\). Using the rule for multiplying two negative numbers, we know:- Negative times negative gives a positive: \(-7 \times -2 = 14\).Understanding this rule helps avoid common mistakes with the signs of the product.

Evaluation of Expressions

Once you have substituted and multiplied correctly, it’s time to evaluate the expression by performing the addition or subtraction. This final step combines all the results to give the value of the expression. In this exercise, after substitution, we reached:- Expression: \(14 + 4\)The expression is clear-cut now, as it only involves addition. Simply add the numbers together:

• Addition is straightforward: \(14 + 4 = 18\).

Evaluating expressions effectively means understanding the order of operations and applying them in the correct sequence. By following the steps: substitute, multiply, then add, we've calculated that the expression's value is 18. Practicing these steps helps in developing a robust foundation for solving various algebraic problems.