Problem 56 - Prealgebra

Question

Find the value of each of the following expressions when \(a = -2\). $$-9 a+3$$

Step-by-Step Solution

Answer

The expression evaluates to 21 when \(a = -2\).

Step 1: Understand the Expression

The given expression is \[-9a + 3\] where \(a\) is a variable. We need to substitute \(a = -2\) and evaluate the expression.

Step 2: Substitute the Value of a

Replace \(a\) in the expression with -2. The expression then becomes:\[-9(-2) + 3\]

Step 3: Multiply the Coefficient by the Substituted Value

Calculate the product of -9 and -2: \[-9 \times -2 = 18\] This means replacing the multiplication in the expression \[-9(-2) + 3 = 18 + 3\]

Step 4: Add the Numbers Together

Now add 18 and 3:\[18 + 3 = 21\] Thus, the expression evaluates to 21 when \(a = -2\).

Key Concepts

Substitution

Substitution is a straightforward process used in algebra to replace variables with their given values. In this exercise, the variable is represented by the letter \( a \). We are provided with the specific value \( a = -2 \). Substitution involves replacing this variable in the algebraic expression with \(-2\). This step is crucial as it allows us to transform a general expression into a specific numerical expression that can be easily evaluated.
For example, in the expression \(-9a + 3\), replacing \( a \) with \(-2\) gives us \(-9(-2) + 3\). This substitution step sets the stage for evaluating the expression using basic arithmetic operations.

Variables

Variables are symbols used in algebra to represent unknown or general numbers. They are usually denoted by letters such as \(a\), \(b\), \(x\), or \(y\). Understanding how variables work is key to solving algebraic expressions.
In the context of the given expression \(-9a + 3\), \(a\) is the variable. This means it's a placeholder that can take on various numerical values. In algebra, the goal is often to evaluate an expression for a given value of the variable, as we've done here with \(a = -2\).
This concept helps us to generalize mathematical formulas and principles, and it underscores the flexibility of algebra in handling different problem scenarios. By substituting different values into the variable \(a\), one can observe how the outcomes change, demonstrating the foundational role variables play in algebra.

Arithmetic Operations

Arithmetic operations are fundamental mathematical operations that include addition, subtraction, multiplication, and division. These operations are integral in evaluating algebraic expressions once variables are substituted. Let's break down the arithmetic operations used in this problem:

• Multiplication: After substituting \(a = -2\), the expression becomes \(-9(-2) + 3\). Here, \(-9\) and \(-2\) are multiplied. Understanding how to multiply negative numbers is essential; multiplying two negatives results in a positive number, giving us \(18\).
• Addition: After multiplication, the expression simplifies to \(18 + 3\). The next step involves performing addition to combine these numbers. Addition is one of the simplest operations, and in this expression, it results in \(21\).

By following these steps and performing the arithmetic operations correctly, we can accurately evaluate any algebraic expression for given values. This emphasizes the importance of these fundamental operations in all aspects of mathematics.