Problem 18 - College Algebra

Question

Let \(x\) represent the number. Use the given conditions to write an equation. Solve the equation and find the number. Eight subtracted from six times a number is \(184 .\) Find the number.

Step-by-Step Solution

Answer

The number is 32.

Step 1: Translate Word Problem Into an Equation

From the statement, 'Eight subtracted from six times a number is 184,' building the equation becomes: \(6x - 8 = 184\) where \(x\) is the number we are trying to find.

Step 2: Solve for the unknown number \(x\)

In order to solve for \(x\), you need to isolate \(x\) on one side of the equation. Begin by adding \(8\) to both sides to undo the subtraction: \(6x = 184 + 8\). Simplifying, you get: \(6x = 192\).

Step 3: Finishing the Solution

The next step is to divide both sides by \(6\). When we do this, the equation becomes: \(x = 192 / 6\). Thus, \(x = 32\).

Key Concepts

Understanding Word Problems

Word problems can initially seem daunting, but breaking them down into smaller parts makes them manageable. A word problem describes a real-life situation using words, which has to be translated into mathematical form.
The process involves identifying the quantities involved and understanding how they relate to each other. In our exercise, the problem states: *Eight subtracted from six times a number is 184.* This involves:

• 'Six times a number' - which suggests multiplication.
• 'Eight subtracted from' - indicating subtraction from the preceding quantity.
• 'Is 184' - implies that the result of the operation equals 184.

By recognizing these elements, you can construct an equation that represents the problem mathematically, making it solvable using the tools of algebra.

Solving with Linear Equations

A linear equation is an equation where the highest power of the variable is one. This makes it straightforward to solve.
In the exercise, the linear equation formed is \(6x - 8 = 184\). Here, the variable \(x\) represents the unknown number we are solving for. Solving it involves:

• Reversing operations to isolate the variable. Begin by adding the same value to each side, which reverses the subtraction. In this case, add \(8\) to both sides, resulting in \(6x = 192\).
• Next, reverse the multiplication by dividing both sides of the equation by \(6\), providing the value of \(x\).

These steps ensure that the manipulation of the equation maintains equality, leading to the solution \(x = 32\).
Linear equations, like these, are foundational in algebra as they simplify complex word problems into manageable mathematical expressions.

Breaking Down Algebraic Expressions

At the core of algebra lies the use of algebraic expressions. These are combinations of numbers, variables, and operations that represent a particular value.
In our example, the expression \(6x - 8\) was used. This expression denotes:

• A multiplier of \(6\) applied to the variable \(x\), representing six times some number.
• The number \(8\) subtracted from this product, altering its value.

A well-structured algebraic expression condenses a lot of information into a compact form, making complex problems easier to tackle once translated into this form.
Understanding each component, like coefficients (the \(6\) in \(6x\)) and constants (the \(-8\)), is a significant step in grasping algebra. It transforms the written problem into a solvable equation, leading to the solution for \(x\), the variable in focus.