Problem 20 - Elementary Algebra

Question

Use the distributive property to rewrite each of the following quantities. $$1(x+y)$$

Step-by-Step Solution

Answer

Question: Rewrite the given expression $$1(x+y)$$ using the distributive property. Answer: The rewritten expression using the distributive property is $$x+y$$.

Step 1: Identify the numbers and variables in the expression

In the given expression $$1(x+y)$$, we have the numbers and variables: a=1, b=x, and c=y.

Step 2: Apply the distributive property

We now apply the distributive property by multiplying the value of 'a' with both 'b' and 'c' and then adding the results: $$1(x+y) = 1\cdot x + 1\cdot y$$

Step 3: Simplify the expression

Let's simplify the expression to get our final answer by multiplying 1 by both x and y: $$1 \cdot x + 1 \cdot y = x+y$$ The rewritten expression using the distributive property is $$x+y$$.

Key Concepts

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating these symbols. It allows us to express mathematical relationships without needing specific numbers. In algebra, variables take the place of numbers so we can solve general problems, rather than specific numerical ones.

• The concept of an equation in algebra is like a balance scale. What is on one side should equal what is on the other.
• Algebra involves operations such as addition, subtraction, multiplication, and division, but it also includes working with expressions, equations, and inequalities.

By learning algebra, you develop the skills to think logically and solve a wide variety of real-world problems.

Expression Simplification

Expression simplification involves reducing an expression into its simplest form. This process makes complex problems more manageable, allowing us to solve them more efficiently.
When simplifying, you rely on various algebraic properties, the distributive property being one of them. It helps rewrite expressions so they're easier to work with.

• A simplified expression should have all like terms combined. For example, in the expression \(1 \cdot x + 1 \cdot y\), the terms with the multiplier 1 can be simplified to \(x + y\).
• Reducing an expression doesn't change its value but helps in easily seeing what the expression represents or finding solutions faster.

Simplifying expressions is a foundational skill in algebra aiding in clear and concise calculations.

Variables

Variables in algebra function as placeholders for values that can change. They are typically represented by letters such as \(x\), \(y\), or \(z\). Understanding variables is critical because they allow us to represent general expressions and solve equations.

• Variables let us represent unknown quantities and model real-world situations. For instance, \(x\) and \(y\) in the expression \(1(x+y)\) are variables representing unknown values.
• When working with variables, you can think of them as containers that hold different values at different times, making algebra flexible and powerful.

Grasping the concept of variables equips you to approach abstract mathematical problems with confidence and competence.