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Chapter 5: Problem 345

In the following exercises, translate and solve. The difference of \(n\) and 1.9 is \(3.4 .\)

Short Answer

Expert verified
The value of is 5.3.

Step by step solution

01

Identify the given information

The problem states that the difference between a number () and 1.9 is equal to 3.4.
02

Write the equation

Translate the given information into a mathematical equation: - 1.9 = 3.4.
03

Isolate the variable

To solve for , add 1.9 to both sides of the equation: - 1.9 + 1.9 = 3.4 + 1.9.
04

Simplify the equation

Simplify both sides of the equation to find the value of : = 5.3.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Translating Word Problems
When faced with a word problem, the first step is to translate everyday language into mathematical expressions. This means identifying numbers, operations, and variables mentioned in the problem. In the example provided, we're told that 'the difference of n and 1.9 is 3.4.'
Here's how we can break that down:
  • 'Difference' refers to subtraction.
  • 'n' is an unknown number, which we can represent as a variable.
  • 'Is' translates to an equals sign (=).
  • '1.9' and '3.4' are numbers mentioned directly in the problem.

Combining these clues, we form the equation: - 1.9 = 3.4. This translation is crucial for solving the problem accurately.
Isolating Variables
After forming the equation, the next step is to isolate the variable鈥攅ssentially getting the unknown number alone on one side of the equation. In our example, we need to solve for in the equation - 1.9 = 3.4.
To isolate , we perform operations that will help us remove other terms from its side of the equation. Since is subtracted by 1.9, we can add 1.9 to both sides of the equation to cancel it out. This gives us:
- 1.9 + 1.9 = 3.4 + 1.9.
Which simplifies to:
= 5.3.
By isolating the variable, we've found that equals 5.3.
Simplifying Equations
Simplifying equations involves performing basic mathematical operations to make the equation easier to understand and solve. Once we've added 1.9 to both sides, we need to make sure each side of the equation is fully simplified.
Let's break it down step by step:
  • On the left side: - 1.9 + 1.9 simplifies to just .
  • On the right side: 3.4 + 1.9 simplifies to 5.3.
Putting it all together, our simplified equation is: = 5.3.
Basic Algebra
Understanding basic algebra is key to solving linear equations like the one in our example. The core principles involve:
  • Operations鈥攁ddition, subtraction, multiplication, and division.
  • Variables are symbols that represent unknown numbers.
  • Equations are mathematical statements showing equality between two expressions.
In the given problem, we've used subtraction to represent 'difference' and addition to isolate the variable. With practice, these steps become more intuitive, and solving similar problems gets easier. Algebra is like a puzzle; each piece (or operation) helps complete the bigger picture.

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