Problem 13 - Algebra and Trigonometry

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: Identify the Type of Equation

Observe whether the given equation is linear or not. A linear equation can be written in the form \(ax + b = c\). Our equation \(2x + 7 = 31\) fits this description, so it is a linear equation.

Step 2: Isolate the Variable Term

To isolate the variable term \(2x\), subtract 7 from both sides of the equation:\(2x + 7 - 7 = 31 - 7\)This simplifies to:\(2x = 24\).

Step 3: Solve for the Variable

Now, solve for \(x\) by dividing both sides of the equation by 2:\(\frac{2x}{2} = \frac{24}{2}\)This simplifies to:\(x = 12\).

Step 4: Verify the Solution

Substitute \(x = 12\) back into the original equation to verify the solution:\(2(12) + 7 = 31\)This simplifies to:\(24 + 7 = 31\).Since both sides are equal, the solution \(x = 12\) is correct.

Key Concepts

Solving Linear Equations

A linear equation is a foundational concept in algebra. It is an equation that, when graphed, forms a straight line. Linear equations are typically in the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants. To solve these equations, the main goal is to find the value of the variable that makes the equation true.
When we encounter a linear equation, like \( 2x + 7 = 31 \), we start by ensuring that we're dealing with a linear format. From there, our task is to perform operations that simplify the equation and isolate the variable on one side, a process which we'll delve into further below. The solution to the equation is the value of \( x \) that satisfies the equation.

Isolating the Variable

Isolating the variable is a critical step in solving linear equations. The objective is to have the variable by itself on one side of the equation. This is achieved through inverse operations, which are operations that "undo" each other.
Here's how we isolate the variable in our example:

• Start with the equation \( 2x + 7 = 31 \).
• Notice the term \( +7 \). Since it is an addition, you subtract 7 from both sides to maintain equality. This simplifies to \( 2x = 24 \).
• Next, the variable \( x \) is multiplied by 2. To isolate \( x \), divide both sides of the equation by 2. This yields \( x = 12 \).

The key is to apply these inverse operations across all terms and ensure that you maintain the balance of the equation. By isolating the variable, you make it possible to identify its exact value.

Verifying Solutions

Verification is an essential last step in solving any equation. It ensures accuracy in your solution and helps catch errors.
Once a solution for the linear equation \( 2x + 7 = 31 \) has been found (i.e., \( x = 12 \)), it's important to substitute this back into the original equation to check if it holds true:

• Replace \( x \) with 12 in the original equation: \( 2(12) + 7 = 31 \).
• Calculate the left side, resulting in 24 + 7, which is 31.
• Since the left side equals the right side, \( x = 12 \) is verified as correct.

Verifying your solution is like double-checking your work to ensure it makes sense. It's a quick step that conclusively proves the correctness of your solution.