Problem 6 - Prealgebra

Question

Define a variable. Then write an equation and solve. A number increased by 8 is 23

Step-by-Step Solution

Answer

The number is 15.

Step 1: Define the Variable

Let's define the variable that represents the unknown number. We'll call this unknown number \( x \).

Step 2: Write the Equation

According to the problem, a number increased by 8 equals 23. In terms of our variable \( x \), this is represented by the equation: \( x + 8 = 23 \).

Step 3: Solve the Equation

To solve for \( x \), we need to isolate it on one side of the equation. We do this by subtracting 8 from both sides: \( x + 8 - 8 = 23 - 8 \). This simplifies to \( x = 15 \).

Step 4: Verify the Solution

To verify our solution, substitute \( x = 15 \) back into the original equation: \( 15 + 8 \). This equals 23, which is correct. Therefore, the solution \( x = 15 \) is verified as correct.

Key Concepts

Variable Definition

Defining a variable is a crucial first step in solving any equation. A variable is simply a symbol, often represented by letters like \( x \) or \( y \), used to stand in for an unknown value that we're trying to find. In the exercise, our task is to determine an unknown number. So, we chose a symbol, \( x \), to act as a placeholder for this number. The power of defining a variable lies in its ability to simplify and structure a math problem.

• The variable \( x \) helps create an expression or equation that represents a real-life problem in a mathematical form.
• By using the variable, we turn words into a format that can be manipulated using algebraic rules.

Without variables, mathematics would lack the precision and efficiency to deal with complex problems, as they provide a universal language to express different scenarios mathematically.

Step-by-Step Solution

Solving an equation like the one in our exercise requires breaking down the problem into manageable steps. First, we take the expression described in words, "a number increased by 8 is 23," and translate it into a mathematical equation. By defining our variable \( x \) as the unknown number, we can express this relationship as \( x + 8 = 23 \). Next, we solve the equation for \( x \) by isolating the variable on one side.

• To isolate \( x \), subtract 8 from both sides of the equation: \( x + 8 - 8 = 23 - 8 \).
• This simplification gives us \( x = 15 \), indicating that the unknown number is 15.

This structured approach ensures clarity and minimizes the potential for errors, as each step logically follows from the one before it. Carefully following these steps will help in solving more complex algebraic problems as well.

Verifying Solutions

Verifying your solution is a vital part of problem-solving in mathematics. Once you have found a solution, it's important to ensure that it truly satisfies the original equation.In our example, we found that \( x = 15 \). To verify, we substitute \( x = 15 \) back into the original equation \( x + 8 = 23 \).

• Substituting gives \( 15 + 8 \), which simplifies to 23.
• Since this matches the right side of the original equation, our solution is confirmed as correct.

Verification helps in identifying any potential mistakes made during the solving process. It's like double-checking your work to make sure everything adds up correctly. Furthermore, mastering verification reinforces understanding and boosts confidence in dealing with more challenging mathematical equations.