Problem 4 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the given equation. $$ 2 x+5=11 $$

Step-by-Step Solution

Answer

The short answer is: To solve the equation \(2x + 5 = 11\), subtract 5 from both sides to obtain \(2x = 6\), then divide both sides by 2 to find \(x = 3\).

Step 1: Subtract 5 from both sides

To isolate the variable term, we need to get rid of the constant term (+5) on the left side. We can do this by subtracting 5 from both sides of the equation: \(2x + 5 - 5 = 11 - 5\) Simplifying, we have: \(2x = 6\) Now we move on to step 2.

Step 2: Divide both sides by 2

To isolate the variable \(x\), we need to get rid of the coefficient (2) in front of the variable. We can do this by dividing both sides of the equation by 2: \(\frac{2x}{2} = \frac{6}{2}\) Simplifying, we have: \(x = 3\) So, the solution to the given equation \(2x + 5 = 11\) is \(x = 3\).

Key Concepts

Variable Isolation

In solving linear equations, one of the primary goals is to isolate the variable. This means getting the variable on one side of the equation by itself with a coefficient of 1. It essentially allows us to "solve for" the unknown. For the equation \(2x + 5 = 11\), we start this process by moving terms around so that \(x\) stands alone on one side. In the first step, we subtract \(5\) from both sides. This helps remove constants, which can be numbers or other known terms, that are added or subtracted from the variable. By isolating the variable, we make it possible to determine its value by performing straightforward mathematical operations.

Inverse Operations

Inverse operations are fundamental tools in solving equations. They are operations that reverse the effect of another operation. For instance, addition is the inverse of subtraction and vice versa. Similarly, multiplication and division are inverse operations of each other. In our exercise, when we encounter \(2x + 5 = 11\), we use inverse operations to simplify and solve it.

• First, we remove the \(+5\) by subtracting \(5\) from both sides, utilizing subtraction as the inverse of addition.
• Next, after simplifying to \(2x = 6\), we use division, the inverse of multiplication, to eliminate the coefficient \(2\) of \(x\) by dividing both sides by \(2\).

By applying inverse operations methodically, we systematically unravel the equation to find the variable’s value.

Simplifying Equations

Simplifying equations involves reducing them to simpler forms while maintaining equality. This step often focuses on reducing the complexity so that the equation is easy to interpret and solve. Once inverse operations are applied, the equation in its simpler form emerges. From our problem, starting with \(2x + 5 = 11\), the equation is simplified in two major parts:

• Subtracting \(5\) results in a more streamlined equation: \(2x = 6\).
• Lastly, dividing the entire equation by \(2\) achieved our simplest form: \(x = 3\).

Simplifying is essential because it reduces possible errors in calculation and provides a clear path to easily solving for the desired variable.