Problem 5 - Algebra Form and Function

Question

In Exercises \(5-8,\) find the sequence of operations to undo the sequence given. Multiply by 5 and then subtract \(2 .\)

Step-by-Step Solution

Answer

Answer: The inverse operations to undo the given sequence are adding 2 and dividing by 5. They should be performed in the reverse order: first, add 2 and then divide by 5.

Step 1: Identify the Inverse Operations

The inverse operations for the given operations are: 1. Add 2: This is the inverse operation of subtract 2. 2. Divide by 5: This is the inverse operation of multiply by 5.

Step 2: Perform Inverse Operations in Reverse Order

To undo the given operations, we perform the inverse operations in reverse order. That means we should first add 2 and then divide by 5. Let's say the original number is x and the result after the given operations is y. We have: \(y = 5x - 2\) Now, we perform the inverse operations: \(x' = \frac{(y + 2)}{5}\) Here, x' is the outcome of undoing the given operations, and it should be equal to the original number x. So, the sequence of operations to undo the given sequence is: add 2 and then divide by 5.

Key Concepts

Algebraic Operations

Algebraic operations are the basic mathematical procedures applied to variables and numbers in expressions and equations. These include addition, subtraction, multiplication, and division. In our exercise, we're dealing with a sequence involving multiplication and subtraction.
When handling these operations:

• Addition and subtraction are inverse operations. This means if you subtract a number, you can add the same number back to undo the effect.
• Multiplication and division are also inverse operations. Dividing by a number is the same as multiplying by its reciprocal.

Knowing these relationships helps us solve equations by using inverse operations to isolate variables. Understanding and manipulating algebraic operations are foundational for solving equations, as they allow us to reshape and solve for the unknown values.

Solving Equations

Solving equations involves finding the value of the unknown that makes the equation true. In our given problem, we're asked to reverse certain operations to retrieve the original number. Here, the equation can be expressed as:
\( y = 5x - 2 \)
This equation represents the sequence of multiplying by 5 and subtracting 2 from an unknown number \( x \). To find \( x \), we solve this equation by reversing the operations:

• First, we undo the subtraction by adding 2 to \( y \).
• Then, we undo the multiplication by dividing by 5.

Solving this yields:
\( x = \frac{y + 2}{5} \)
These reversed steps provide the solution for \( x \). Techniques like these showcase the power of inverse operations in solving linear equations.

Sequence of Operations

In mathematics, the sequence of operations refers to the order in which different actions are performed to solve an expression or problem. The original sequence provided was to multiply by 5 and subtract 2. Understanding the sequence is vital because it determines how numbers are manipulated.
When reversing or undoing a sequence of operations, it’s crucial to perform the inverse operations in the opposite order:

• Identify all operations involved.
• Determine their inverse operations.
• Apply these inverse operations in reverse sequence.

For our problem:

• Start with subtracting 2, so add 2 as the inverse.
• Next, the multiply by 5 operation is undone by dividing by 5.

This approach ensures that the effect of each original operation is completely nullified, thereby retrieving the initial value before the sequence began. It's like rewinding a mathematical process step by step.