Problem 29 - Fundamentals of Mathematics

Question

Determine the missing factor(s). \(38=2 .\) ______.

Step-by-Step Solution

Answer

The missing factor is 19.

Step 1: Understand the Equation

We are given the equation \(38 = 2 \times \text{____}\) and need to find the value that makes this equation true.

Step 2: Isolate the Missing Factor

To isolate the missing factor, we can rewrite the equation as \(38 = 2 \times x\). This means we need to solve for \(x\).

Step 3: Solve for the Missing Factor

To find \(x\), divide both sides of the equation by 2. The equation becomes \(x = \frac{38}{2}\).

Step 4: Perform the Division

Carry out the division \(\frac{38}{2}\) which simplifies to 19. Therefore, \(x = 19\).

Key Concepts

Missing Factor

When dealing with equations, a **missing factor** is an unknown value that completes a multiplication statement. In our exercise, the equation is given as \( 38 = 2 \times \text{____} \). Here, the blank represents the missing factor we need to determine. Finding a missing factor is a common problem in algebra where one of the numbers in a multiplication operation is unknown. This hidden value is what you solve for to make the equation true.

• Think of the problem like a puzzle: out of the pieces given, one piece (our missing factor) is hidden.
• We label this piece as \( x \) in our equation to signify an unknown.
• By discovering \( x \), you complete the multiplication equation.

Getting comfortable with identifying and solving for a missing factor is crucial for mastering algebraic concepts.

Equation Solving

**Equation solving** is all about finding the unknown value that balances an equation. In the exercise, we have the equation \( 38 = 2 \times x \). Through solving, we want to zero in on that unknown (the missing factor) which in this case is \( x \).

• Think of an equation as a scale; each side should weigh the same to keep it balanced.
• Our task is to manipulate or "solve" the equation to determine what \( x \) must be to make this true.

Solving an equation often involves performing operations that "undo" what is done to the unknown. This balancing act of ensuring both sides stay equal is the heart of equation solving.

Division

**Division** is a mathematical operation that is crucial for isolating unknowns in equations. In the context of our exercise, division helps separate the missing factor (\( x \)) from the constant multiplier (2). To solve \( 38 = 2 \times x \), we divide both sides by 2.

• Division "cancels out" the number that is attached to the missing factor.
• This leaves the factor on one side of the equation and a simplified number on the other side.
• In this case, dividing gives us \( x = \frac{38}{2} \).

Understanding division as the inverse of multiplication is crucial. Just as multiplication increases the value, division breaks it down to find its base components. Mastering this will make solving similar equations much easier.

Isolation of Variable

**Isolation of a variable** refers to the process of rearranging an algebraic equation in such a way that the unknown variable is by itself on one side of the equation. In our scenario, the variable \( x \) is paired with 2. To isolate \( x \), we undo the multiplication by dividing both sides by 2.

• The goal is to have the equation in the form \( x = \text{some value} \).
• This provides a clear understanding of what \( x \) equals.

A well-isolated variable tells us exactly what that unknown value is without any additional operations muddying the result. In this case, after isolation, \( x \) equals 19, giving us a clear and direct answer. This process is fundamental in solving more complex algebraic equations, where multiple steps may be involved to isolate the variable.