Problem 8 - Elementary Algebra

Question

For the following problems, the first quantity represents the product and the second quantity a factor. Find the other factor. $$ 5 x+25,5 $$

Step-by-Step Solution

Answer

Question: Find the other factor if the product is 5x + 25 and one of the factors is 5. Answer: The other factor is x + 5.

Step 1: Rewrite the problem as a multiplication

Given the product is \(5x + 25\) and one of the factors is 5, so we rewrite the product with 5 as a factor: $$(5x + 25) = 5 \cdot F$$ Where \(F\) is the other factor we are looking for.

Step 2: Divide by the given factor

Now, we will simply divide the product by the given factor of 5 to find the other factor, \(F\): $$F = \frac{5x + 25}{5}$$

Step 3: Simplify the expression

To find the other factor, \(F\), simplify the expression by dividing both terms by 5: $$F = \frac{5x}{5} + \frac{25}{5}$$

Step 4: Calculate the final expression

On simplifying the terms, we get: $$F = x + 5$$ So, the other factor we were looking for is \(x + 5\).

Key Concepts

Elementary Algebra

Elementary algebra is the branch of mathematics that deals with the basic concepts of algebra including variables, expressions, equations, and the rules of operations that apply to numbers, mostly focusing on real numbers. It is often considered the stepping stone for advanced mathematics and is a fundamental aspect of a student's mathematical education. It is in elementary algebra that students learn to manipulate expressions and understand the properties of algebraic structures.

In our exercise, the elementary algebra concept is demonstrated through the factoring of an algebraic expression, which is the process of breaking down a complex expression into simpler 'factors' that, when multiplied together, will result in the original expression. Knowing how to factor is crucial as it simplifies many algebra problems and is a key skill in solving equations.

Simplifying Expressions

The process of simplifying expressions in algebra involves reducing them to a simpler or more efficient form while retaining their original value. This often includes combining like terms, factoring, and employing basic arithmetic operations such as addition, subtraction, multiplication, and division.

In the context of the given problem, simplifying the expression involves dividing the algebraic expression by one of its factors. Here, both terms in the expression share a common factor, which can be divided out to simplify the expression. This process makes it easier to understand and work with algebraic expressions, since it presents them in their most reduced form. It also helps in solving equations and inequalities when the expressions are as simple as possible.

Multiplication and Division

Multiplication and division are two of the basic arithmetic operations that are extensively used in algebra. In the world of algebra, these operations are not only performed with numbers but also with variables and algebraic expressions. Multiplication involves the combining of quantities, whereas division is the process of distributing a quantity into equal parts.

In our step-by-step solution, we employ division to find the unknown factor of an algebraic expression. The expression is divided by a known factor (in this case, 5), which further simplifies the problem by reducing it to an even more elementary form. Understanding how to properly apply these operations to algebraic expressions is critical, as they serve as tools to rearrange and simplify expressions or equations, leading to the extraction of important information like variable values or other factors.