Problem 52 - Elementary Algebra

Question

For the following problems, the first quantity represents the product and the second quantity a factor. Find the other factor. $$ a-b+c,-1 $$

Step-by-Step Solution

Answer

Question: Given that the product is (a-b+c) and one of its factors is (-1), find the other factor. Answer: The other factor is -a+b-c.

Step 1: Write down the product and known factor

We are given the product \((a-b+c)\) and a factor \((-1)\).

Step 2: Calculate the other factor

We need to find the other factor, let's call it \(x\), such that \((a-b+c) = (-1) \cdot x\). To find \(x\), we simply divide the product by the known factor: $$ x = \frac{(a-b+c)}{-1} $$

Step 3: Simplify the expression

Now, we will simplify the expression by dividing each term by \((-1)\): $$ x = -a + b - c $$ Therefore, the other factor is \(-a+b-c\).

Key Concepts

Algebraic Expressions

Understanding algebra starts with grasping the concept of an algebraic expression. These expressions are combinations of letters and numbers interconnected through mathematical operations such as addition, subtraction, multiplication, and division. The letters, often referred to as variables, represent unknown values that we aim to find.

For instance, in the equation \(a-b+c\), \(a\), \(b\), and \(c\) are variables. This equation itself is an algebraic expression where \(b\) is subtracted from \(a\), and then \(c\) is added to the result. Algebraic expressions can manifest in various forms and complexities, from simple linear expressions to more complex polynomial equations. The fundamental idea is that by manipulating these expressions using algebraic laws and properties, we can solve for the unknown variables and understand the relationships between different mathematical quantities.

When dealing with expressions, the ability to interpret and simplify them becomes vital. This leads us naturally to the concept of factors and products, which are integral parts of algebraic expressions.

Factors and Products

The terms 'factor' and 'product' are fundamental in multiplication operations within algebra. A product is the result of multiplying factors together. For example, if we multiply 2 and 3, we get 6, so we say that 6 is the product of 2 and 3, and that 2 and 3 are factors of 6.

In algebra, factors are not just numbers but can also include variables or whole expressions. In the exercise \(a-b+c\), if multiplied by \(x\), where \(x\) represents another factor, the product is still \(a-b+c\). Here, we see expressions acting as factors, illustrating that the principles of multiplication extend beyond numerical boundaries to encompass algebraic expressions. This slightly more abstract concept of factors and products paves the way for simplifying algebraic expressions, enabling students to solve equations efficiently.

Simplifying Expressions

The process of simplifying expressions is crucial for solving algebraic equations and understanding algebraic relationships. Simplification makes expressions easier to work with by reducing them to their simplest form. To simplify an expression, one combines like terms, uses the distributive property, and cancels out terms where possible.

In our exercise, simplifying involves finding the other factor of the product \(a-b+c\) given the known factor \( -1\). By dividing the product by the known factor, the other factor is isolated, resulting in \(x = -a+b-c\). This simplification allows us to clearly see the relationship between the product and its factors—each term in the product \(a-b+c\) is simply multiplied by \( -1\) to find the other factor \(x\).

This efficient approach to simplification is a fundamental skill in algebra that aids students in reducing complex problems into simpler ones, ultimately leading to a clearer understanding and the correct solution.