Problem 16 - College Algebra

Question

Let \(x\) represent the number. Use the given conditions to write an equation. Solve the equation and find the number. The sum of a number and 29 is \(54 .\) Find the number.

Step-by-Step Solution

Answer

The unknown number \(x\) is 25.

Step 1: Define the Equation

From the problem, the sum of a number \(x\) and 29 is equal to 54. As such, the equation will be \(x + 29 = 54\). This equation represents the relationship between the unknown number \(x\) and the result 54.

Step 2: Solve for 'x'

To isolate \(x\), the unknown number, subtract 29 from both sides of the equation. This will give \(x + 29 - 29 = 54 -29\). After simplifying, the result is \(x = 25\).

Step 3: Validate the Result

To ensure that \(x = 25\) is the correct solution, substitute \(x\) with 25 in the initial equation. Doing this leads to \(25 + 29 = 54\), which reaffirms that the solution \(x = 25\) is correct.

Key Concepts

Solving Linear Equations

Linear equations are mathematical expressions that show the equality between two expressions. The core goal is to find the value of the unknown variable, often represented as \(x\). In the context of solving an equation like \(x + 29 = 54\), we are trying to determine which number, when added to 29, will give 54.

To solve this type of equation:

• Identify the unknown variable \(x\).
• Isolate \(x\) on one side of the equation by performing arithmetic operations. Here, subtract 29 from both sides: \(x + 29 - 29 = 54 - 29\).
• This helps you find \(x = 25\).

By isolating \(x\), you determine that 25 is the number that satisfies the equation. Remember to perform the same operation on both sides of the equation to maintain its balance.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators (like addition or subtraction). In the given problem, \(x + 29\) is the algebraic expression representing the sum of a number \(x\) and 29. Algebra uses these expressions to generalize mathematical concepts and is essential in forming equations like the one in our problem.

Understanding algebraic expressions involves:

• Identifying constants (e.g., 29 in \(x + 29 = 54\)).
• Recognizing variables (e.g., \(x\)).
• Applying arithmetic operations to form an expression or equation.

In simple terms, algebraic expressions are structured recipes of numbers and symbols that help us model real-world problems mathematically. This structure allows us to logically solve for unknown values.

Validating Solutions

Validating a solution is crucial to ensure that it is correct. Once you solve the equation, you need to "check" your result. For the problem \(x + 29 = 54\), we found \(x = 25\). The validation step involves substituting \(x = 25\) back into the original equation.

During validation:

• Replace the variable in the original expression with the solution: \(25 + 29\).
• Check that both sides of the equation are equal: \(25 + 29 = 54\).
• If the equation holds true, the solution is valid.

This step is critical because it confirms solving was done correctly. If the equation doesn't balance, there's an error in calculation that needs revisiting. Always double-check your work to ensure your solution is accurate.