Problem 60 - Algebra 1: Concepts and Skills

Question

You thought the balance in your checkbook was \(\$ 53,\) but when your bank statement arrived, you realized that you forgot to record a check. The bank statement lists your balance as \(\$ 47 .\) Let \(x\) represent the value of the check that you forgot to record. Which equation is a correct model for the situation? Solve the correct equation. A. \(53-x=47\) B. \(x-47=53\)

Step-by-Step Solution

Answer

The correct equation to model this problem is \(53 - x = 47\). Solving this equation gives \(x = 6 \). So the unrecorded check was for $6.

Step 1: Analyze the problem

There was an initial amount of $53 in the checkbook, and then an unrecorded check was written. This means the balance is reduced by the amount of the check, which is represented by 'x'. Since the remaining balance according to the bank statement is $47, we're looking for an equation that models this subtraction from the initial balance. Therefore, the correct equation that models this problem is '53 - x = 47'.

Step 2: Solve the equation

To solve for 'x', we need to isolate 'x' on one side of the equation. We can do this by adding 'x' to both sides to get rid of the '-x' on the left side: \( 53 - x + x = 47 + x \) This simplifies to \( 53 = 47 + x \) Next, subtract 47 from both sides to isolate 'x': \( 53 - 47 = 47 - 47 + x \) This simplifies to \( x = 6 \)

Key Concepts

Solving Linear Equations

Understanding the principles of solving linear equations is essential for mastering algebra. A linear equation is a mathematical statement where two expressions are set equal to each other, with the expressions typically involving a constant and a variable. The goal is to find the value of the unknown variable that makes the equation true.

To solve a linear equation, several steps are usually followed. A key step is to isolate the variable on one side of the equal sign, which involves using inverse operations such as adding, subtracting, multiplying, or dividing. It's important to perform the same operation on both sides of the equation to maintain equality. For the given example, where the original balance is \(53, and the bank statement reflects a balance of \)47 after an unrecorded check, represented by 'x', the equation reflects a decrease in balance:

Subtract to Isolate the Variable

To isolate 'x', we subtract the final balance from the initial balance \( 53 - x = 47 \). By adding 'x' to both sides to cancel out the subtraction, you simplify the expression to \( 53 = 47 + x \), and then subtract 47 from both sides. Thus, you find that the value of the missed check, 'x', is $6.

Writing Equations from Word Problems

Word problems can be translated into algebraic equations with practice and understanding of the context. Bearing in mind the context, we must define what the variables represent and construct an equation that reflects the scenario described.

In this particular example, a real-world situation involving a checkbook balance is presented. The process of translating the word problem into an equation involves several steps:

Identify the Variables

Begin by determining what 'x' represents, which in this case is the amount of the forgotten check.

Set up the Equation

To reflect the described scenario, we interpret the situation: starting with an initial balance (53 dollars), a check 'x' was forgotten, and the new balance is now 47 dollars. Thus, setting up the equation \( 53 - x = 47 \) captures the essence of the word problem. The skill of writing equations from word problems involves abstract thinking and an ability to convert verbal descriptions into mathematical language.

Algebraic Reasoning

Algebraic reasoning is the process of logically working through mathematical problems to find the relationships between different parts. It involves forming conjectures, validating ideas, and drawing conclusions based on algebraic principles.

In the context of our example, algebraic reasoning allows us to deduce that since the balance decreased by the amount of the check, we must subtract the value of the check from the initial balance to find the final balance. This reasoning leads us to choose the equation \( 53 - x = 47 \) over \( x - 47 = 53 \).

Logic Over Guesswork

Algebraic reasoning transcends trial and error by applying logical rules to arrive at the solution systematically and predictably. It includes understanding the properties of real numbers, the ability to manipulate variables, and the capacity to interpret mathematical statements correctly. This kind of reasoning is a cornerstone for problem-solving in a variety of fields, opening doors to advanced mathematics and real-world applications alike.