Problem 20 - Prealgebra

Question

Define a variable. Then write an equation and solve. A number minus 10 is 27 .

Step-by-Step Solution

Answer

The number is 37.

Step 1: Define the Variable

Let's define a variable to represent the unknown number. We'll use the variable \( x \) to represent this number.

Step 2: Write the Equation

According to the problem, a number minus 10 is 27. We can express this information as the equation: \( x - 10 = 27 \).

Step 3: Solve the Equation

To find the value of \( x \), add 10 to both sides of the equation: \( x - 10 + 10 = 27 + 10 \). This simplifies to \( x = 37 \).

Key Concepts

Understanding Variables

In mathematics, a variable is a symbol, usually a letter, that represents a number we don't yet know. Think of it as a placeholder in an equation. When trying to solve problems, defining a variable is often the first step. Here, we chose the letter \( x \) as our variable to stand for the unknown number. By assigning a variable, we make it easier to write and manipulate equations. Remember, variables are essential tools in math for expressing the unknown, whether it's a single number or a whole range of possibilities.

The Role of Equations

An equation is like a balance scale, where both sides are equal. The left-hand side and the right-hand side must weigh the same. In our exercise, the equation is \( x - 10 = 27 \). This equation tells us that if you take some number \( x \) and subtract 10, you will get 27.
Writing equations involves translating words into math. This process includes identifying operations like addition or subtraction and recognizing how to express relationships mathematically.
Equations are powerful because they can represent many different types of problems and allow us to solve them systematically.

Prealgebra Essentials

Prealgebra is the foundation for all higher math topics. It involves understanding and manipulating basic mathematical concepts, including arithmetic and simple algebraic expressions. Concepts such as variables and equations are key.
In prealgebra, we learn to translate everyday problems into mathematical expressions and use operations to find solutions. Being comfortable with these skills sets the stage for more complex problem-solving tasks in algebra, geometry, and beyond.

Steps for Effective Problem Solving

Problem solving in math is like solving a puzzle. It involves understanding the problem, devising a plan, carrying out that plan, and checking the result. In our example:

• First, define what you're looking for using a variable.
• Next, translate what the problem says into an equation.
• Then, use arithmetic to solve the equation step by step.
• Finally, verify your solution by substituting back into the original problem.

By following a structured approach, you can tackle even the trickiest math problems with confidence. Always ensure to review each step to minimize errors and deepen your understanding.