Problem 56 - Algebra 2

Question

Simplify each expression. 5\((4 x+2 y-x+2)\)

Step-by-Step Solution

Answer

The simplified expression is \(15x + 10y + 10\).

Step 1: Distribute the constant

Begin by distributing the constant \(5\) to each term inside the parentheses. This is done by multiplying \(5\) with each of the terms within the parentheses: \[5(4x + 2y - x + 2) = (5 imes 4x) + (5 imes 2y) + (5 imes -x) + (5 imes 2)\]This calculation results in:\[20x + 10y - 5x + 10\]

Step 2: Combine like terms

Now combine the like terms. Like terms are terms that contain the same variables raised to the same power. In the expression, \(20x - 5x\) are like terms.Apply the operation to combine the coefficients of \(x\):\[20x - 5x = 15x\]Thus, the simplified expression becomes:\[15x + 10y + 10\]

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that is used to simplify expressions by removing parentheses. It states that when you multiply a number by a sum, it is the same as multiplying the number by each addend separately and then adding the products. For example, for any numbers a, b, and c, the distributive property is expressed as:

• \( a(b + c) = ab + ac \)

In our exercise, we used this property to handle the expression \(5(4x + 2y - x + 2)\). We multiplied each term inside the parentheses by 5:

• \( 5 \times 4x = 20x \)
• \( 5 \times 2y = 10y \)
• \( 5 \times -x = -5x \)
• \( 5 \times 2 = 10 \)

After applying the distributive property, we obtained \(20x + 10y - 5x + 10\). This shows how the distributive property helps distribute the multiplication over the terms within the parentheses.

Combining Like Terms

When simplifying an algebraic expression, combining like terms is an essential step that makes the expression more concise and easier to understand. "Like terms" are terms in an expression that have the same variable raised to the same power. The coefficients of these terms can be added or subtracted, while their variables remain unchanged.From our example, after distributing, we had the expression \(20x + 10y - 5x + 10\). Here:

• \(20x\) and \(-5x\) are like terms because they both contain the variable \(x\) to the first power.
• To combine them, add their coefficients: \(20 - 5 = 15\).
• This results in \(15x\).
• The terms \(10y\) and \(10\) do not change because they don't have like terms with the same variables or without variables, respectively.

Thus, the simplified expression becomes \(15x + 10y + 10\). Combining like terms creates a cleaner and more manageable algebraic expression.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They can range from simple to complex, but all follow specific rules that guide their simplification and manipulation.In our exercise, the expression we started with was \(5(4x + 2y - x + 2)\), a typical example of an algebraic expression. Simplifying such expressions involves understanding several key components:

• Constants: These are fixed numbers, like 2 or 5 in the original expression.
• Variables: Symbols representing unknown numbers, such as \(x\) and \(y\) in this problem.
• Coefficients: Numbers that multiply the variables, like 4 in \(4x\) or -1 in \(-x\).

The process often involves applying the distributive property and combining like terms to simplify into a shorter, equivalent expression. Understanding algebraic expressions is vital since they're used to model real-world situations, solve equations, and perform in-depth mathematical analysis.