Problem 9 - Algebra and Trigonometry

Question

\(9- 46\) 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 to the equation is \(x = 12\).

Step 1: Understanding the Equation

The equation given is \(2x + 7 = 31\). This is a linear equation because it can be written in the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants and \(x\) is the variable.

Step 2: Isolating the Variable Term

To isolate the term with \(x\), subtract 7 from both sides of the equation. This gives us:\[2x + 7 - 7 = 31 - 7\]Simplifying both sides leads to:\[2x = 24\]

Step 3: Solving for x

To solve for \(x\), divide both sides of the equation by 2:\[\frac{2x}{2} = \frac{24}{2}\]This simplifies to:\[x = 12\]

Step 4: Verifying the Solution

Substitute \(x = 12\) back into the original equation to verify the solution is correct:\[2(12) + 7 = 31\]Simplifies to:\[24 + 7 = 31\]Since both sides of the equation are equal, our solution \(x = 12\) is verified.

Key Concepts

Solving Equations

Understanding how to solve equations is like finding the missing piece in a puzzle. The goal is to determine the value of the variable, often referred to as 'x', that satisfies the given equation. In a linear equation, such as \(2x + 7 = 31\), this means finding the value of \(x\) that makes both sides of the equation equal.
The crucial steps involved are:

• Manipulating both sides of the equation: You can add, subtract, multiply, or divide both sides by the same non-zero number without changing the equation's solution.
• Goal of simplification: Simplify the equation step by step to get the variable \(x\) alone on one side of the equation.

The process involves moving around numbers and coefficients until you can cleanly solve for \(x\). Each modification takes you closer to revealing that elusive variable.

Isolation of Variable

The method of isolation of the variable is a cornerstone technique in solving equations. This means you rearrange the equation until the variable stands alone on one side. For our equation \(2x + 7 = 31\), the isolation process unfolds as follows:

• Eliminate constants alongside \(x\): Remove constants from one side by performing the inverse operation. In this case, subtract 7 from both sides to keep the equation balanced, leading to \(2x = 24\).
• Clear the coefficient of \(x\): Coefficients are the numbers directly multiplied by the variable. Here, divide both sides by 2 to achieve \(x = 12\).

Each step you take to simplify brings us closer to isolating \(x\), ensuring that eventually, the variable stands alone, ready to reveal its value.

Verifying Solutions

After finding a solution, it's always wise to verify that your solution is correct. Verifying solutions means checking your work by substituting the value back into the original equation. This final step ensures that your solution truly satisfies the equation. Let's look at the example of solving \(2x + 7 = 31\) with the solution \(x = 12\).
Verify by substituting:

• Replace \(x\) in the original equation with 12.
• Calculate \(2 \cdot 12 + 7\) which simplifies to \(24 + 7 = 31\).
• Ensure that this is equal to the other side of the equation, which indeed is 31.

If both sides match, you can be confident that \(x = 12\) is the correct solution. Verification builds trust in your solving method and helps you ensure accuracy.