Problem 3 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(1-20,\) let \(x\) represent the mumber. Use the given conditions to write an equation. Solve the equation and find the mumber. A number decreased by 23 is equal to \(214 .\) Find the number.

Step-by-Step Solution

Answer

The unknown number is 237.

Step 1: Formulation of the equation

Based on the phrasing of the problem, an equation can be constructed as \(x - 23 = 214\), where \(x\) represents the unknown number.

Step 2: Solving the equation

To isolate \(x\) and thus find the value of the unknown number, 23 can be added to both sides of the equation yielding \(x = 214 + 23\).

Step 3: Final calculation

Performing the addition on the right side of the equation yields that \(x = 237\).

Key Concepts

Solving Equations

When tackling algebraic equations, the ability to solve for an unknown variable is a fundamental skill. To solve an equation, such as the one presented in the given exercise (\(x - 23 = 214\)), the goal is to isolate the variable, making it the subject of the formula.

In this scenario, we aim to unravel what the original number was before it decreased by 23. To do this, we perform the inverse operation of subtraction, which is addition. By adding 23 to both sides of the equation, the balance of the equation is maintained and we are able to determine the true value of the unknown variable. Hence, the equation simplifies to \(x = 214 + 23\), leading us to the solution of \(x = 237\). It is essential to execute operations consistently on both sides of an equation, as maintaining equality is pivotal when solving.

Equation Formulation

The process of creating an algebraic representation of a word problem is known as equation formulation. Understanding the relationship between the quantities involved is the essence of this step.

In our problem, we are given a situation: 'A number decreased by 23 is equal to 214.' The language here clearly indicates a subtraction operation. Therefore, we represent the unknown number by a variable, commonly denoted as \(x\), and then translate the sentence into the equation \(x - 23 = 214\). This formulation turns a wordy description into a concise mathematical expression that we can solve. Proper translation from words to numbers and operations is integral in algebra problem-solving and helps set the stage for finding the correct solution.

Algebra Problem Solving

Approaching an algebra problem methodically can help simplify the process and ensure accuracy. First, you need to understand the problem and identify what is being asked.

Next, devise a plan by determining which mathematical operations are needed. For the given problem, we formulated the equation based on the conditions stated. Upon establishing your plan, which is the equation \(x - 23 = 214\), you execute it by solving for \(x\), yielding \(x = 237\). Lastly, verify the solution by plugging it back into the original equation to see if it holds true. For further practice, you can try similar problems, like solving for a number increased by a certain value, or double-checking that when you decrease your solution by 23, you really do get 214. By following these steps, you develop a strategy that can be applied to a broad range of algebra problems.