Problem 65 - Algebra 2

Question

Solve each equation. $$ 2 x+11=25 $$

Step-by-Step Solution

Answer

The solution is \(x = 7\).

Step 1: Subtract 11 from both sides

To start solving the equation \(2x + 11 = 25\), we first need to isolate the term with \(x\). We do this by subtracting 11 from both sides of the equation, so we have: \(2x + 11 - 11 = 25 - 11\). Simplifying gives us \(2x = 14\).

Step 2: Divide both sides by 2

Now that we have \(2x = 14\), we will divide each side by 2 to solve for \(x\). This yields \(x = \frac{14}{2}\), which simplifies to \(x = 7\).

Step 3: Conclusion

Therefore, the solution to the equation \(2x + 11 = 25\) is \(x = 7\).

Key Concepts

Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. It allows us to represent real-world problems in mathematical terms. Learning algebra is crucial as it forms the foundation for advanced studies in mathematics and science.
In our example, the equation given is linear, which means it has variables raised only to the power of one. Linear equations like this one typically have a straightforward method for finding solutions.

• Algebra helps us understand relationships between numbers, making it a very practical tool for problem-solving.
• It covers various topics, including solving equations, manipulating expressions, and understanding functions.

The goal is to find the value of the variable that makes the equation true. Understanding these foundational concepts will help you tackle more complex problems in the future.

Equation Solving

Equation solving involves finding the values of variables that make an equation true. It's an essential part of algebra that deals with equations like our example, \(2x + 11 = 25\). Solving equations often involves manipulating the equation to isolate the variable on one side.
Let's break down the process:

• Identify the term that contains the variable. In our case, it's \(2x\).
• Perform operations to both sides of the equation to maintain balance. We subtract 11 from both sides to simplify to \(2x = 14\).
• Finalize the isolation of \(x\) by dividing both sides by 2, resulting in \(x = 7\).

These steps help ensure that each operation is performed with the goal of isolating the variable, a crucial technique in solving any equation.

Mathematics Education

Mathematics education focuses on equipping learners with the skills and understanding necessary to engage with mathematical concepts effectively. When we solve problems like linear equations, it's essential to grasp the underlying methods.
Teaching equation solving in stages, such as in the provided solution, helps to develop a methodical approach:

• It is important to encourage students to ask questions and explore different methods of solving equations.
• Emphasize understanding over memorization, ensuring students comprehend why each step is necessary.
• Relate mathematical concepts to real-life situations to make learning more attractive and relatable.

Providing a solid foundation helps students to succeed not only in mathematics but also enhances critical thinking and problem-solving skills applicable in everyday life.