Problem 7 - Elementary Algebra

Question

Simplify. (Assume all denominators are nonzero.) $$ \begin{array}{ll} 1-32 & 54-13 \end{array} $$

Step-by-Step Solution

Answer

-31 and 41

Step 1: Simplify the First Expression

Take the expression to the left, \(1 - 32\), and perform the subtraction. Subtract 32 from 1 to simplify the expression: \[1 - 32 = -31\]

Step 2: Simplify the Second Expression

Now, take the expression to the right, \(54 - 13\), and perform the subtraction. Subtract 13 from 54 to simplify the expression: \[54 - 13 = 41\]

Step 3: Combine the Simplified Expressions

Combine the simplified results from both expressions. The final result is \(-31\) and \(41\) respectively.

Key Concepts

Subtraction

Subtraction is one of the basic operations in mathematics and forms the foundation for more complex calculations. It involves taking one number away from another. For example, in the expression \(1 - 32\), the number 32 is subtracted from 1.
To effectively perform subtraction, follow these steps:

• Identify the minuend (the number from which another number is subtracted). In \(1 - 32\), 1 is the minuend.
• Identify the subtrahend (the number that is being subtracted). Here, 32 is the subtrahend.
• Subtract the subtrahend from the minuend to find the difference.

Subtraction is helpful in determining how much more or less one quantity is compared to another. It is essential in algebraic simplification, where simplifying expressions often involves subtracting terms.

Negative Numbers

Negative numbers are numbers less than zero and are represented with a minus sign (-). When performing subtraction, sometimes the result is a negative number.
For example, in the expression \(1 - 32 = -31\), the result is -31 because 32 is greater than 1, thus leaving a deficit of 31. Negative numbers can be thought of as moving to the left on the number line or having less of a quantity.

• Understand that subtracting a higher number from a lower number results in a negative.
• Negative numbers are integral to operations involving debts, temperature below zero, and sea levels below the surface.

Working with negative numbers might seem tricky initially, but with practice, it becomes quite intuitive and straightforward.

Step-by-Step Solutions

Step-by-step solutions are methods of breaking down problems into smaller, manageable parts to solve them effectively. This approach is beneficial in understanding algebraic simplification tasks.
In the given exercise:

• First step: Simplify each expression. For \(1 - 32\), compute \(-31\) and for \(54 - 13\), compute \(41\).
• Next step: Combine the results which have been found separately.

Each step focuses on a specific part of the problem, allowing students to concentrate on one piece at a time, making complex problems easier to understand. This method is especially helpful for learners who are beginning to explore more advanced mathematics.