Problem 34 - Fundamentals of Mathematics

Question

Translate each phrase or sentence to a mathematical expression or equation. When eight times a number is increased by five, the result is equal to the original number plus twenty-six.

Step-by-Step Solution

Answer

The unknown number is 3.

Step 1: Identify the Variables

Define the variable to represent the unknown number. Let's use \( x \) to represent the unknown number. So, \( x = \text{the unknown number} \).

Step 2: Translate Each Part of the Phrase

Translate 'eight times a number' as \( 8x \). Then translate 'increased by five' as adding 5 to this expression, giving us \( 8x + 5 \).

Step 3: Establish the Equation

The phrase 'the result is equal to the original number plus twenty-six' translates to an equation. It means \( 8x + 5 = x + 26 \).

Step 4: Rearrange the Equation

To solve for \( x \), start by subtracting \( x \) from both sides: \( 8x - x + 5 = 26 \). This simplifies to \( 7x + 5 = 26 \).

Step 5: Solve for the Unknown Variable

Subtract 5 from both sides to isolate terms with \( x \): \( 7x + 5 - 5 = 26 - 5 \). This simplifies to \( 7x = 21 \).

Step 6: Find the Value of x

Finally, divide both sides by 7 to solve for \( x \): \( x = \frac{21}{7} \). So, \( x = 3 \).

Key Concepts

Equation Solving

In mathematics, solving equations involves finding the value of the unknown variable that makes the equation true. An equation typically presents a mathematical problem using an equal sign, which indicates that two expressions are equivalent. To solve the equation, follow these steps:

• Understand the equation and what is being asked. For example, in the equation from the original exercise, both sides need to represent the same value once solved.
• Start by isolating the variable, which often involves moving terms around and performing mathematical operations, such as addition, subtraction, multiplication, or division, on both sides of the equation.
• Verify your solution by substituting the value back into the original equation to ensure that both sides are equal. For instance, with the solution of the original equation, substituting back confirms the result as correct: \( 8(3) + 5 = 3 + 26 \) simplifies to \( 29 = 29 \).

Equation solving is a crucial skill and forms the foundation for many advanced mathematical problems. Practice regularly to improve your confidence in this area.

Variable Identification

A variable in mathematics is a symbol used to represent an unknown value or quantity. Usually denoted by letters such as \( x \), \( y \), or \( z \), variables are a fundamental part of forming equations. Identifying which variable to use involves:

• Reading the problem carefully and determining what is unknown. For the original exercise, the unknown number referred to as 'a number' is represented by \( x \).
• Ensuring consistency in the use of the variable throughout the equation-solving process. This helps maintain clarity as you work through mathematical expressions and steps.
• Assigning specific variables when multiple unknowns are involved. Each should clearly represent a different unknown quantity, allowing for effective equation setup and solution.

Successfully identifying and defining variables is key, as they allow us to translate everyday language problems into mathematical expressions, forming the first step in solving them.

Phrase Translation

Phrase translation involves converting written language into mathematical language, transforming words into symbols and equations. This is a common task in algebra that involves several steps:

• Break down the sentence and identify mathematical operations, such as 'times,' 'plus,' and 'increased by,' which are analogous to multiplication, addition, and other arithmetic operations.
• Translate each part of the phrase systematically. For instance, 'eight times a number' becomes \( 8x \), and 'increased by five' translates to adding 5, forming \( 8x + 5 \).
• Combine translated expressions to form a complete equation. 'The result is equal to the original number plus twenty-six' becomes \( 8x + 5 = x + 26 \). This step is crucial for setting up the problem correctly for solution.

Phrase translation helps bridge common language and mathematical expressions, enabling us to solve real-world problems systematically. Mastering this skill enhances problem-solving efficiency and comprehension.