Problem 118 - Algebra and Trigonometry

Question

Exercises \(116-118\) will help you prepare for the material covered in the next section. Simplify: \(4(5 x+4 y)-2(6 x-9 y)\)

Step-by-Step Solution

Answer

The simplified form of the given expression is \(8x + 34y\)

Step 1: Distribute the factors

First, distribute the factors within the parentheses on each of the given terms: \(4(5x) + 4(4y) - 2(6x) + 2(9y) = 20x + 16y - 12x + 18y\). Here the distributive property is applied, which states that for all real numbers a, b, c, a(b + c) = ab + ac, and a(b - c) = ab - ac.

Step 2: Combine like terms

Now, combine like terms on the expression: \(20x - 12x + 16y + 18y = 8x + 34y\). This is done by adding or subtracting the coefficients of the same variable.

Key Concepts

Distributive Property

Understanding the distributive property is essential when simplifying algebraic expressions. It's a rule that lets us multiply a single term and two or more terms inside a set of parentheses. To put it simply, if you have an expression like \( a(b + c) \), you can 'distribute' the \( a \) across the \( b \) and \( c \) which then gives you \( ab + ac \). This property also applies to subtraction: \( a(b - c) = ab - ac \).

In the exercise \( 4(5 x + 4 y) - 2(6 x - 9 y) \), the distributive property is used to expand the expression. The number outside the parenthesis is multiplied with each term inside the parenthesis, separately. This operation helps in breaking down complex expressions into simpler, more manageable pieces for further simplification.

Combining Like Terms

Once you've distributed your terms, the next step in simplifying an algebraic expression is to combine like terms. Like terms are terms that have the same variable raised to the same power. Only the coefficients of these terms are different. For example, \( 2x \) and \( 3x \) are like terms because they both have the variable \( x \), but \( 2x \) and \( 3y \) are not like terms because they have different variables.

In our problem, after distributing we get \( 20x + 16y - 12x + 18y \). We then combine like terms by adding or subtracting their coefficients. This step is crucial for simplification and leaves us with a more condensed version of the expression. So here, \( 20x - 12x \) becomes \( 8x \), and \( 16y + 18y \) becomes \( 34y \), leading to our simplified result: \( 8x + 34y \).

Algebraic Operations

Algebraic operations refer to the basic mathematical processes used to manipulate algebraic expressions, including addition, subtraction, multiplication, and division of polynomials. When simplifying expressions, it's important to perform these operations correctly while strictly following the order of operations, commonly remembered with the acronym PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction).

For instance, if an expression involved dividing like terms, you would divide their coefficients and keep the variable part intact. In the expression given in the exercise, multiplication is the key algebraic operation used initially (through the distributive property), followed by addition and subtraction when combining like terms. Fully understanding each operation and when to apply them facilitates the simplification process and helps in solving more complex algebraic equations.