Problem 56 - Elementary Algebra

Question

For the following problems, use the distributive property to expand the quantities. $$x(2 y+5)$$

Step-by-Step Solution

Answer

Answer: The expanded form of the expression is \(2xy + 5x\).

Step 1: Identify the distributive property

We have to apply the distributive property of multiplication on the expression \(x(2y+5)\). Distributive property states that \(a(b+c) = ab + ac\).

Step 2: Apply the distributive property

Using the distributive property, we will expand the given expression as follows: $$x(2y + 5) = x(2y) + x(5)$$

Step 3: Simplify the expression

Now, multiply x with each term inside the parentheses: $$x(2y) = 2xy$$ $$x(5) = 5x$$ So, the simplified expression is: $$2xy + 5x$$ The expanded expression using the distributive property is \(2xy + 5x\).

Key Concepts

Algebra

Algebra is the branch of mathematics where we use symbols and letters to represent numbers and quantities in mathematical expressions and equations. These symbols allow us to formulate general rules about numbers and the arithmetic operations that link them.
Understanding these rules and how to manipulate these symbols is key to solving algebraic problems. Let's break down two important concepts in algebra to give you a clearer understanding:

• Variables: Represent unknown or changeable values. In our expression, both "x" and "y" are variables.
• Coefficients: Numbers that multiply the variables. In the term "2y", "2" is the coefficient of "y".

When solving algebraic problems, learning to maneuver around variables and coefficients is essential. By forming equations using these symbols, we can apply rules like the distributive property, making it easier to handle more complex expressions and solve problems efficiently.

Expansion of Expressions

Expansion of expressions involves rewriting an expression in an extended form. It's a crucial part of algebra that often uses the distributive property to simplify expressions. By expanding expressions, we can make them more manageable and solve equations efficiently.
Consider the expression we worked on: \(x(2y + 5)\). The idea is to distribute the variable or number outside the parentheses to each term inside, effectively expanding the expression. Here's how it happens:

• The "x" is multiplied by "2y" to become \(2xy\), extending the term.
• Then "x" multiplies "5", resulting in \(5x\).

The new expression, \(2xy + 5x\), is the expanded form. Using expansion helps refashion expressions in a way that's more straightforward and easier to understand, especially when aiming to isolate and solve for variables in more complicated equations.

Simplification of Expressions

Simplification makes algebraic expressions clearer and less complex by combining like terms or making the expression easier to work with. Often, after expanding an expression, the next step is to simplify it, which helps in solving equations quicker.
For the expression \(x(2y + 5)\) expanded to \(2xy + 5x\), simplification can involve several actions:

• Combine like terms if possible, although our current example doesn't allow this as \(2xy\) and \(5x\) are distinctly different terms.
• Look for patterns or factors that could simplify the expression further in more complicated equations.

Ultimately, the goal of simplification is to convert any expression into its simplest, most easily understandable form. A simplified expression provides a clearer path to finding the solution to algebraic equations, guiding you closer to finding what values your variables represent.