Problem 53 - Prealgebra

Question

Find the value of each of the following expressions when \(a = -2\). $$2 a+5$$

Step-by-Step Solution

Answer

The value of the expression is 1.

Step 1: Substitute the given value

Substitute the given value of \(a\) into the expression. We are given \(a = -2\). So, replace \(a\) with \(-2\) in the expression: \(2a + 5\).

Step 2: Multiply the constant with the variable

Multiply the constant \(2\) with \(-2\). This will be \(2 \times (-2)\). This calculation is \(-4\).

Step 3: Add the constant term

Add the constant term \(5\) to the result of the multiplication from Step 2. So, perform the operation \(-4 + 5\).

Step 4: Calculate the final result

Compute the addition from Step 3: \(-4 + 5 = 1\). This is the value of the expression.

Key Concepts

Substitution in Expressions

Substitution is a straightforward but important concept when evaluating expressions. When you're given an algebraic expression and a specific value for the variable, substitution is the process where you replace the variable with that value. It's like solving a puzzle by filling in the missing pieces. For example, in the expression \(2a + 5\), if you're told that \(a = -2\), you simply replace every instance of \(a\) with \(-2\). This helps in converting an algebraic expression into a numerical one, making it easier to work further calculations on.

• Look for the variable in the expression.
• Replace the variable with the given number.
• Make sure to accurately use the given values without mistake.

Once you substitute correctly, you move on to the next step which involves simplifying the expression.

Simplifying Expressions

Simplifying expressions is about breaking them down into the simplest form. After substitution, your expression is no longer in algebraic form but becomes numerical. The goal is to perform arithmetic operations in a standard order. This process ensures that you've made the expression as simple as possible to obtain a straightforward solution. For \(2a + 5\) with \(a = -2\), once \(a\) is substituted, the expression becomes \(2(-2) + 5\). Simplification here isn't just about making the expression smaller but about preparing it for the arithmetic operations that follow.

• Perform any multiplication or division as dictated by the terms in the expression.
• Reorganize the expression by performing parts in brackets or any indicated operations first.
• Order of operations matters; remember to resolve from left to right and follow the BIDMAS/BODMAS rules (Brackets, Indices, Division/Multiplication, Addition/Subtraction).

By simplifying expressions correctly, you pave the path for a more accurate and quick calculation of the final result.

Arithmetic Operations

Arithmetic operations involve basic math functions like addition, subtraction, multiplication, and division. These are the building blocks of evaluating expressions. Once you've substituted and simplified your expression, you're ready to finally compute the result using arithmetic. In the example of the expression \(2(-2) + 5\), the first operation you'll do is multiplication. Here, multiplying 2 by \(-2\) gives you \(-4\). Subsequent steps involve handling operations like addition or subtraction.

• Add or subtract terms as needed to simplify.
• Keep track of positive and negative signs throughout.
• Make use of a calculator for more complicated calculations to ensure accuracy.

In the given expression, after multiplication, you perform addition: sum up \(-4 + 5\) which easily leads you to the answer, which is 1. The structured use of arithmetic operations ensures that mathematical problems can be approached step by step, eliminating errors often found in calculations.