Problem 43 - Algebra and Trigonometry

Question

\(35-54\) . Perform the addition or subtraction and simplify. $$ u+1+\frac{u}{u+1} $$

Step-by-Step Solution

Answer

The result of the subtraction is \(-19\).

Step 1: Identify the operation

We are tasked with the subtraction of two numbers: 35 and 54. Our goal is to simplify the expression by performing this subtraction.

Step 2: Perform the subtraction

To find the result of the subtraction, compute \( 35 - 54 \). Since 35 is smaller than 54, the result will be negative.

Step 3: Simplify the result

Subtract 54 from 35, which gives \( 35 - 54 = -19 \). The subtraction simplifies to \(-19\).

Key Concepts

Expression Simplification

Expression simplification involves breaking down complex expressions into their simplest form. This process helps in making equations easier to solve by reducing redundancy and complexity. Simplifying expressions requires understanding and exploiting algebraic properties, such as combining like terms and factoring. For example, if you encounter an expression like \(x + 3x + 2\), you can simplify it by combining like terms (those that have the same variable raised to the same power). Thus, \(x + 3x + 2\) simplifies to \(4x + 2\).

Often, expressions contain fractions or terms with denominators, which require common denominators for further simplification. Simplifying rational expressions also includes canceling out common factors from the numerator and the denominator. Keeping these steps in mind is crucial for dealing with different forms of algebraic expressions.

Subtraction of Integers

Subtraction of integers is a fundamental arithmetic operation that involves finding the difference between two numbers. When subtracting integers, it is helpful to understand the directional movement on a number line. Starting from the first number, move left if you are subtracting, which often results in a lower or negative number. Consider our example operation, \(35 - 54\).

To compute this by hand:

• Start at 35 on the number line.
• Move 54 steps to the left, passing zero into the negative territory.

Ultimately, this leads to an answer of \(-19\).

When dealing with larger numbers or negative values, always ensure that you keep track of the signs. A common mistake is losing a negative sign during operations, so double-check your work to ensure accuracy.

Negative Numbers

Negative numbers are numbers less than zero and play a crucial role in mathematics. They are used to represent values on the opposite side of the number line, such as debt, temperature below zero, or elevations below sea level. When working with negative numbers, the direction and operation determine the outcome of mathematical problems.

For subtraction, incorporating negative numbers adds an extra layer of complexity. Consider the expression \(35 - 54\). Since 35 is a smaller number than 54, subtracting 54 will result in a negative outcome, which is \(-19\). Keep these thoughts in mind when dealing with negatives:

• A negative number subtracted from a positive often results in a more negative number.
• The order of operation is vital; reversing a subtraction can lead to a different result, \(54 - 35 = 19\), not \(-19\).
• Practice moving along the number line to the left when subtracting and moving right when adding.

Understanding and correctly applying operations with negative numbers is crucial for solving real-world problems and handling equations efficiently.