Problem 10 - Elementary Algebra

Question

In the following problems, the first quantity represents the product and the second quantity represents a factor of that product. Find the other factor. $$ 30 y^{4}, 6 $$

Step-by-Step Solution

Answer

Answer: The other factor is 5y^4.

Step 1: 1. Write down the given product and factor

We are given a product: $$30y^4$$, and one of its factors: $$6$$.

Step 2: 2. Write the equation representing the problem

Let's denote the missing factor as x. The equation representing the given problem can be written as: $$30y^4 = 6x$$.

Step 3: 3. Solve the equation for x

To find the missing factor x, we need to solve the above equation for x. We can do this by dividing both sides of the equation by 6: $$x = \frac{30y^4}{6}$$.

Step 4: 4. Simplify the expression for x

We can simplify the expression for x by dividing 30 by 6: $$x = 5y^4$$.

Step 5: 5. State the other factor

The other factor of the given product (30y^4) is $$5y^4$$.

Key Concepts

Multiplication

Multiplication is a fundamental arithmetic operation that involves combining quantities. When we talk about multiplication in algebra, it implies finding the product of numbers, variables, or a combination of both. Consider

• Multiplying numbers such as 6 and 5 gives us 30. This process is straightforward when dealing with numerical values.
• Multiplaying variables involves combining their factors. For instance, multiplying two like terms such as \(y^2\) and \(y^3\) gives us \(y^{2+3} = y^5\). Here, the exponents are added.

In this exercise, the product we deal with is expressed in terms of both numbers and variables, specifically \(30y^4\). To solve for a missing factor, one must understand how both numbers and variables are involved in multiplication. It provides the foundational knowledge needed for more complex algebraic manipulations. In particular, knowing how to reverse the operation (using division) is crucial when finding a missing factor.

Factoring

Factoring involves breaking down a mathematical expression into its components, known as factors. These components, when multiplied together, are equivalent to the original expression. In essence, it is the reverse process of multiplication. Let's delve deeper:

• For numerical expressions, factoring means finding numbers that multiply together to form the original number. For example, with 30, factors can be 5 and 6, since \(5 \times 6 = 30\).
• In algebraic expressions, factoring may include variables and exponents, as showcased in \(30y^4 = 6 \times 5y^4\). Here, the terms 6 and \(5y^4\) multiply to form the original product.

By recognizing different ways to factor an expression, one can simplify complex problems, solve equations efficiently, and understand the structure of algebraic expressions better. Factoring is a vital skill for simplifying and solving problems in algebra, enhancing one’s ability to tackle algebraic challenges.

Simplifying expressions

Simplifying expressions in algebra means transforming complex expressions into simpler forms while preserving their value. This process is essential in making equations easier to understand and solve. Here’s how you can simplify an expression:

• Start by identifying terms you can divide out. For instance, in the equation \(30y^4 = 6x\), you divide both sides by 6 to simplify the expression for \(x\).
• Perform operations such as division to reduce coefficients. For example, \( \frac{30}{6} = 5\).
• If variables are involved, apply rules of exponents. Since there's only power of \(y^4\) in this problem, no further simplification in the exponent is necessary.

Simplifying ensures that expressions are in their simplest form, making them easier to interpret and work with. Mastering this skill is crucial as it forms the groundwork for more advanced algebraic manipulations and solving equations effectively.