Problem 11 - Introductory and Intermediate Algebra for College Students 4th

Question

Let \(x\) represent the number. Use the given conditions to write an equation. Solve the equation and find the number. Seven subtracted from five times a number is \(178 .\) Find the number.

Step-by-Step Solution

Answer

The number represented by 'x' is 37.

Step 1: Write the equation

According to the given conditions, the number represented by variable 'x' is calculated by subtracting 7 from 5 times the number. Therefore, the equivalent equation is \(5x - 7 = 178\).

Step 2: Solve for 'x' by adding 7 to both sides

To isolate the term with 'x' on one side of the equation, add 7 to both sides of the equation: \(5x = 178 + 7\). The equation then becomes \(5x = 185\).

Step 3: Final calculation to get the number 'x'

To find the value of the variable 'x', divide both sides of the equation by 5: \(x = 185 / 5\). After division, we find that \(x = 37\).Therefore, the number being sought is 37.

Key Concepts

Variable Representation

In algebra, a variable is a symbol used to represent an unknown value. This symbol is often a letter, such as \(x\), that stands in place of the number we are trying to find.
Variables are essential as they allow us to form equations that model real-world situations. By using a variable, we can succinctly express complex problems mathematically.

• In the given exercise, the problem involves finding the number for which the equation "seven subtracted from five times a number is 178" holds true.
• Here, \(x\) is used to represent the unknown number. Thus, all operations and calculations will revolve around this variable.

Choosing the right variable representation helps in setting up the equation easily, making it possible to apply mathematical rules to solve the problem systematically.

Solving Linear Equations

Solving linear equations is a foundational skill in algebra that involves finding the value of the unknown variable that satisfies the equation. A linear equation is an equation of the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.

To solve a linear equation effectively, follow these steps:

• Identify the terms involving the variable and the constant terms.
• Use inverse operations to isolate the variable on one side of the equation. This often involves adding, subtracting, multiplying, or dividing both sides by the same number.
• Perform the calculations step-by-step to simplify the equation.

In our exercise, the given equation was initially \(5x - 7 = 178\). To solve for \(x\), we added 7 to both sides, obtaining \(5x = 185\). Then, we divided each side by 5 to find \(x = 37\).

Remember, maintaining a systematic approach can prevent mistakes and lead directly to the solution.

Step-by-Step Solutions

A step-by-step solution breaks down the process of solving a problem into manageable pieces, making it easier to understand each part of the procedure.

The advantage of this approach is that it provides a clear pathway from the given problem to the final answer, highlighting the reasoning behind each step.

In the exercise, we followed a sequence of operations:

• Firstly, we formed the equation \(5x - 7 = 178\) based on the problem statement.
• Next, to isolate \(x\), we added 7 to both sides, simplifying the equation to \(5x = 185\).
• Finally, by dividing both sides by 5, we found that \(x = 37\).

The step-by-step process aids in understanding not just how to perform calculations, but also why each action is taken. This clarity is crucial in building a strong foundation for tackling more complex algebraic problems in the future.