Problem 63 - Elementary Algebra

Question

For the following problems, use the distributive property to expand the quantities. $$z(x+y+m)$$

Step-by-Step Solution

Answer

Question: Expand the following expression using the distributive property: z(x + y + m) Answer: zx + zy + zm

Step 1: Identify the multiplication

Here, z is the factor outside the parentheses that needs to be multiplied with each term inside the parentheses (x, y, and m).

Step 2: Apply the distributive property

To expand the expression, multiply z with each term inside the parentheses. Using the distributive property, we get: $$z(x + y + m) = zx + zy + zm$$ The expression has now been expanded using the distributive property.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (such as addition, subtraction, multiplication, and division). In our example, the expression z(x+y+m) is an algebraic expression. Here's why:

• z, x, y, and m are variables representing quantities that may vary.
• (x+y+m) shows addition within the parentheses.
• The combination of these variables with the addition operation forms a group of terms that can be manipulated algebraically.

The beauty of algebraic expressions is that they can be simplified or expanded to express the relationships between variables in different ways, which is particularly useful when solving equations or inequalities.

Expanding Expressions

Expanding expressions is a core skill in algebra that involves rewriting an expression in an equivalent form with all products carried out. For instance, the expanded form of z(x+y+m) shows the distributive property in action:

z(x + y + m) = zx + zy + zm

This means that we distributed the multiplication of z over each term within the parentheses. It's essential to ensure each term is individually multiplied by z, not just one or two of them. This process makes it easier to see all the terms that are part of the expression and to further manipulate or simplify the expression as needed for solving an equation or understanding a formula.

Why Expand Expressions?

Expanding can be particularly useful when solving equations, as it allows you to isolate variables and combine like terms. It also paves the way for factoring, which is essentially the reverse process of expanding.

Elementary Algebra

Elementary algebra is the branch of mathematics that deals with the simple properties of numbers and the basic operations performed on them. This includes manipulating algebraic expressions, solving linear equations, and understanding functions and graphs. The distributive property, used to expand expressions as seen in our example, is a fundamental aspect of elementary algebra.

In elementary algebra, symbols and letters are used to represent numbers. This allows us to solve problems without yet knowing the exact values. By learning and understanding elementary algebra, you gain foundational skills that are essential for advanced mathematical concepts and a wide range of real-world applications from science to finance.