Problem 15 - College Algebra

Question

Let \(x\) represent the number. Use the given conditions to write an equation. Solve the equation and find the number. A number increased by 40 is equal to \(450 .\) Find the number.

Step-by-Step Solution

Answer

The number in question is 410.

Step 1: Formulate the equation

From the description, we know that when a number is increased by 40, it equals 450. This can be written into an equation: \(x + 40 = 450\), where 'x' is the number we're looking for.

Step 2: Solve for 'x'

To find the value of 'x', subtract 40 from both sides of the equation: \(x = 450 - 40\).

Step 3: Calculate 'x'

Subtracting 40 from 450 gives 410. So the equation simplifies to: \(x = 410\). This means the number we're looking for is 410.

Step 4: Validate the solution

To validate the solution, substitute 'x' with 410 in the original equation: \(410 + 40 = 450\). As expected, the left-hand side equals the right-hand side, validating our solution.

Key Concepts

Algebraic Equations

Algebraic equations are a fundamental part of algebra and mathematics in general. These equations involve variables and constants, often expressed with operations like addition, subtraction, multiplication, and division. In simple terms, they are mathematical sentences that show the relationship between different values.

Understanding algebraic equations is crucial because they allow us to solve for unknown values. In the exercise provided, we have an algebraic equation of the form \( x + 40 = 450 \). Here, "x" represents an unknown number, while 40 and 450 are constants that help describe the relationship between the numbers.

• Variables: The unknown values we're solving for, denoted by letters like x, y, or z.
• Constants: Known values that remain fixed, such as 40 and 450 in our example.
• Operations: Basic arithmetic operations like addition, subtraction, involved in forming the equation.

By solving these equations, we reveal the hidden values represented by the variables.

Equation Formulation

Equation formulation is the process of representing a real-world problem as an algebraic equation. Before solving, it's important to correctly translate the words of a problem into a mathematical expression.

For example, in the problem, "A number increased by 40 is equal to 450," the task is to determine the number. Step one is to identify the unknown, which is the number itself. Let this unknown be denoted by \( x \).

• Identify the unknown: What is the problem asking you to find? In this case, it's the number represented by \( x \).
• Translate the statement: Convert the words into an equation. "Increased by 40" translates to addition, resulting in \( x + 40 \).
• Set the equality: The equation must reflect that it equals 450, which completes the equation \( x + 40 = 450 \).

This clear and careful translation process is crucial for solving the equation accurately.

Problem-Solving Steps

Problem-solving steps are key methods used to find solutions to equations. Approaching systematically helps avoid errors and ensures that you're on the right track to finding the solution.

When solving the equation \( x + 40 = 450 \), you should:

• Identify the operation needed: Determine which operation will help isolate \( x \). In this case, it's subtraction to remove 40 from the left side.
• Perform the operation: Subtract 40 from both sides to keep the equation balanced, resulting in \( x = 450 - 40 \).
• Simplify to find the unknown: Calculate the result to find \( x = 410 \).
• Check your work: Substitute \( x \) back into the original equation to verify, \( 410 + 40 = 450 \), confirming the solution is correct.

Following these steps provides a reliable path to problem-solving in algebra.

Basic Arithmetic Operations

Understanding basic arithmetic operations is essential when solving algebraic equations. These operations include addition, subtraction, multiplication, and division, and they form the backbone of manipulating equations.

In the example problem, the essential operation focused on is subtraction. You need to subtract 40 from 450 to solve for \( x \).

• Addition: Combining values. Initially used in the problem to "increase by 40."
• Subtraction: Removing values. Used to isolate \( x \) by subtracting 40 from 450.
• Multiplication and Division: Although not used in this particular problem, they are also key operations in solving different types of algebraic equations.

Grasping these basic operations enables you to effectively manipulate and solve equations.