Problem 67 - Elementary Algebra

Question

Translate the phrases or sentences to mathematical expressions or equations. A number is divided by twice the number, and eight times the number is added to that result. The result is negative one.

Step-by-Step Solution

Answer

Question: Translate the phrases into a mathematical equation and solve: "A number is divided by twice the number, and eight times the number is added to that result. The result is -1." Answer: The equation is \(\frac{1}{2} + 8x = -1\), and the solution is \(x = -\frac{3}{16}\).

Step 1: Define the Unknown

Let x be the number we want to find.

Step 2: Translate the First Phrase

The first phrase is: "a number is divided by twice the number." We can write this expression as \(\frac{x}{2x}\).

Step 3: Translate the Second Phrase

The second phrase is: "eight times the number is added to that result." We add 8 times the number (8x) to the result of the expression from Step 2, which can be written as \(\frac{x}{2x} + 8x\).

Step 4: Form the Equation

According to the problem, the result of the entire expression is equal to negative one. This can be written as the equation: \(\frac{x}{2x} + 8x = -1\).

Step 5: Solve the Equation

Now we need to solve the equation for x: \(\frac{x}{2x} + 8x = -1\) First, we simplify the fraction since both numerator and denominator share a factor of x: \(\frac{x}{2x} = \frac{1}{2}\) So, the equation becomes: \(\frac{1}{2} + 8x = -1\) Now, we isolate the term with x by subtracting \(\frac{1}{2}\) from both sides: \(8x = -1 - \frac{1}{2}\) or \(8x = -\frac{3}{2}\) Finally, we find the value of x by dividing both sides by 8: \(x = \frac{-\frac{3}{2}}{8}\) or \(x = -\frac{3}{16}\) Therefore, the number is \(x = -\frac{3}{16}\).

Key Concepts

Algebraic Expressions

When dealing with problems like this, we transform real-world phrases into algebraic expressions. This is the foundation of translating sentences into mathematical language.
Algebraic expressions consist of numbers, operations, and possibly variables. They describe a mathematical situation or condition.
For example, in the first step of our problem, the phrase "a number is divided by twice the number" is turned into the expression \( \frac{x}{2x} \). Here, \( x \) represents the unknown number.
The main task is to read words and identify the mathematical operations they imply:

• "The number" translates to \( x \).
• "Divided by" suggests a fraction.
• "Twice the number" becomes \( 2x \).

By turning the language of the problem into these mathematical statements, we pave the way for solving the actual problem.

Solving Linear Equations

Solving linear equations involves finding the value of a variable that makes the equation true. These equations take the form \( ax + b = c \), where **x** is the variable, and **a**, **b**, and **c** are numbers.
In our given problem, we have formed the equation \( \frac{1}{2} + 8x = -1 \).
To solve it, we perform a series of logical arithmetic operations to isolate **x**:

• First, we subtract \( \frac{1}{2} \) from both sides to get rid of the constant term on the left.
• This leaves us with \( 8x = -\frac{3}{2} \).
• Finally, we divide by 8, to solve for **x**: \( x = -\frac{3}{16} \).

Each step uses simple addition, subtraction, or division, following the balance principle of equality, ensuring whatever operation we perform on one side, we do the same on the other side. This allows for solving the equation systematically.

Defining Variables

Defining variables is a crucial skill when working with algebraic expressions. Variables are symbols, usually letters, representing unknown values or quantities in equations.
This process begins by identifying what is "unknown" in the problem statement.
In our exercise, "a number" that we want to determine is described as such. To solve the problem, we assign this unknown number a variable, commonly **x**.
By defining variables:

• We clarify what each part of the word problem refers to in mathematical terms.
• Allow mathematical expressions and equations to be simplified and interpreted easily.

This is a fundamental part of translating words into math because it enables us to set up reliable mathematical models based on descriptions given in problems.

Fraction Simplification

Fraction simplification is an essential technique when solving equations involving fractions. This process makes expressions easier to manage and solve.
In our problem, we first dealt with the fraction \( \frac{x}{2x} \). Looking closer, both **x** in the numerator and **2x** in the denominator shared a common factor (**x**) which meant this fraction could be simplified.
Simplification steps usually involve:

• Identifying common factors between the numerator and denominator.
• Dividing these out to achieve a reduced fraction. Here, \( \frac{x}{2x} \) simplifies to \( \frac{1}{2} \).

Through simplification, fractions become less complex, paving the way for more straightforward arithmetic operations, such as addition, subtraction, or solving the resulting equations.