Problem 35 - Algebra 2

Question

Evaluate each expression for \(a=3\) and \(b=-5\) \(2 a+b\)

Step-by-Step Solution

Answer

The value of the expression when evaluated for a=3 and b=-5 is 1.

Step 1: Substitute the given values

Replace the variables 'a' and 'b' with their given values in the expression. So, you replace 'a' with 3 and 'b' with -5 into the expression '2a + b'.

Step 2: Evaluate the expression

After substitution, the expression becomes '2(3) + (-5)'. Multiply 2 with 3 to get 6, and then add -5 to 6.

Step 3: Simplify the result

Simplify the result of the operation to get the final answer. Adding 6 and -5 gives you 1.

Key Concepts

Substitution in Algebra

In algebra, the process of substitution involves replacing variables with their corresponding numerical values. It's like a 'search and replace' function where instead of letters, you get specific numbers. Let's make it clearer using an example: suppose you have the algebraic expression 2a + b and you're given the values a=3 and b=-5. You apply substitution by replacing 'a' with 3 and 'b' with -5. Thus, your original expression 2a + b transforms into 2(3) + (-5).

Substitution is a crucial foundational skill because it allows you to see how equations change as the values of variables change. Moreover, it prepares you for more complex operations like solving systems of equations where you'll often substitute one equation into another to find a solution.

Simplifying Expressions

The goal of simplifying expressions is to make them as easy to understand as possible. After you've substituted numbers for variables, you often get something that can be streamlined. For instance, after substitution, the expression 2(3) + (-5) needs to be simplified.

This process involves two steps: performing the arithmetic operations (in this case, multiplication and addition), and then combining like terms if any are present. Here, you first do the multiplication: 2 multiplied by 3, which gives you 6. Then you add -5 to 6, which simplifies the expression down to 1. Simplification makes expressions cleaner and prepares you to use or analyze them further in equations or inequalities.

Arithmetic Operations

Arithmetic operations are the bread and butter of algebra; they're the basic building blocks that allow us to calculate and simplify expressions. The four primary operations are addition, subtraction, multiplication, and division. In the context of our example, once we've substituted 3 for 'a' and -5 for 'b', we encounter two operations: multiplication and addition.

Let's dissect it: first, we multiply 2 by 3, which is straightforward. Then, we face a slightly trickier concept: adding a negative number, -5, to our result. Adding a negative is essentially the same as subtracting its positive counterpart. Therefore, adding -5 is the same as subtracting 5 from 6, which yields 1. It's vital to understand how to navigate through these basic operations seamlessly as they are integral in solving more complex algebraic problems.