Problem 11 - Precalculus Mathematics for Calculus

Question

The given equation is either linear or equivalent to a linear equation. Solve the equation. $$2 x+7=31$$

Step-by-Step Solution

Answer

The solution is \(x = 12\).

Step 1: Isolate the variable term

To solve the equation \(2x + 7 = 31\), you need to isolate the variable term on one side. Start by subtracting 7 from both sides of the equation:\[2x + 7 - 7 = 31 - 7\]This simplifies to:\[2x = 24\]

Step 2: Solve for the variable

Now that you have the simplified equation \(2x = 24\), solve for \(x\) by dividing both sides by 2:\[\frac{2x}{2} = \frac{24}{2}\]This gives:\[x = 12\]

Key Concepts

Isolating Variables

In the realm of solving linear equations, the concept of isolating the variable is pivotal. This essentially means arranging the equation such that the variable (often called \(x\)) stands alone on one side of the equation. To achieve this, you must perform operations that "undo" other terms. For instance, in the original equation \(2x + 7 = 31\), your goal is to make \(x\) isolated. You start by eliminating constants. Here, you subtract 7 from both sides. Why both sides? Because equations are like a balanced scale; whatever you do to one side, you must do to the other to maintain balance. This subtraction simplifies the equation to \(2x = 24\). Key things to remember when isolating variables:

• Perform the same operation on both sides of the equation.
• Focus on reversing operations (e.g., add to counteract subtraction).
• Your goal is to have the variable alone on one side.

Simplifying Equations

Simplifying equations is another crucial step when dealing with linear equations. Simplification involves performing arithmetic operations to clean up and compress the equation into a more manageable form. After isolating the variable-term, simplify the expression if necessary. In our example, once you subtract 7 from both sides in \(2x + 7 = 31\) and end up with \(2x = 24\), you have already simplified part of the process by removing constants from being added or subtracted. Remember:

• It's useful to simplify step by step — clutter can lead to mistakes.
• Each step should bring you closer to isolating and solving for the variable.
• A simplified equation is easier to solve and understand.

Basic Algebraic Operations

Basic algebraic operations form the backbone of solving linear equations. They encompass addition, subtraction, multiplication, and division. In our specific equation \(2x + 7 = 31\), you've already used subtraction to simplify the left side. The next step involves another operation to isolate \(x\): division. Once the equation is modified to \(2x = 24\), you divide both sides by 2 to solve for \(x\). This operation helps in dissolving the multiplication by 2 impacting \(x\), landing on \(x = 24/2 = 12\). Important pointers:

• Whenever you multiply or divide by a number, apply it to the whole side.
• Division is crucial when negating multiplication alongside variables.
• Remember the primary goal of these operations: revealing the value of the variable.

Mastering these operations provides a toolbox for unraveling any equation you face!