Problem 66 - Elementary Algebra

Question

For the following problems, use the distributive property to expand the quantities. $$1(x+y)$$

Step-by-Step Solution

Answer

Answer: The expanded form of the given expression is \(x + y\).

Step 1: Identify the distributive property

In this case, the distributive property to be used is: $$a(b+c) = ab + ac$$ Here, \(a=1\), \(b=x\), and \(c=y\).

Step 2: Apply the distributive property

Multiply 1 by each term inside the parentheses: $$1(x+y) = 1\cdot x + 1\cdot y$$

Step 3: Simplify the expression

Since multiplying any value by 1 does not change its value, the simplified expression would be: $$1(x+y) = x + y$$

Key Concepts

Algebraic Expression

An algebraic expression is a mathematical phrase that contains numbers, variables—symbols that represent unknown values—and operations such as addition or multiplication. For instance, in the expression \(1(x+y)\), \(1\) is the numerical coefficient, while \(x\) and \(y\) are variables standing for unknown quantities. The expression combines these elements through multiplication and addition.

Understanding algebraic expressions is fundamental because they are used to model real-world situations and solve problems mathematically. These expressions allow us to succinctly write and manipulate mathematical ideas, setting the stage for further exploration and solution in algebra.

Simplifying Expressions

Simplifying expressions is a process in algebra that involves reducing an expression to its most basic form without changing its value. This can involve combining like terms, using the distributive property, and eliminating unnecessary mathematical operations.

For example, to simplify \(1(x+y)\), we distribute the \(1\) across each term inside the parentheses, resulting in \(1\cdot x + 1\cdot y\). Because multiplying by \(1\) does not alter the value of the variable, we can further simplify to \(x + y\). Simplifying makes expressions easier to read and work with, particularly when solving equations or working on more complex algebraic problems.

Elementary Algebra

Elementary algebra is a branch of mathematics that deals with the manipulation and solution of equations involving variables. It lays the groundwork for understanding more advanced topics in mathematics. One of its key tools is the distributive property, which helps in expanding expressions and solving algebraic equations.

In the provided example, utilizing the distributive property \(a(b+c) = ab + ac\) is a core aspect of elementary algebra. This property makes it possible to simplify and transform expressions, such as changing \(1(x+y)\) into \(x + y\). Mastering elementary algebra requires a good grasp of these properties and the ability to apply them in various contexts to simplify and solve algebraic expressions and equations.