Problem 6 - Algebra 2

Question

Simplify. $$ 2 x(3 y+9) $$

Step-by-Step Solution

Answer

The simplified expression is \(6xy + 18x\).

Step 1: Distribute the Coefficient

To simplify the expression, start by distributing the coefficient outside the parentheses, which is 2x, to each term inside the parentheses (3y + 9). This is done by multiplying 2x with 3y, and then multiplying 2x with 9.

Step 2: Multiply the Terms

Multiply each term individually: - First, multiply 2x and 3y to get 6xy. - Then multiply 2x and 9 to get 18x.

Step 3: Combine the Terms

After distributing and multiplying, combine the results to form the simplified expression. So, 6xy and 18x are the simplified terms of the expression.

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential skill in algebra that involves reducing the expression to its simplest form. This process helps make the expression easier to understand and work with for further calculations or evaluations. In our original exercise, we have the expression \(2x(3y + 9)\). Simplifying begins by applying the distributive property, which is a fundamental algebraic principle. This property allows us to distribute the term outside the parentheses—\(2x\), in this case—to each term inside the parentheses.To do this:

• Multiply \(2x\) by \(3y\), resulting in \(6xy\).
• Next, multiply \(2x\) by \(9\), resulting in \(18x\).

Once you have performed these multiplications, the expression becomes \(6xy + 18x\). The expression is now simplified, as no further combinations or reductions can be made.
Simplifying expressions helps in solving algebraic equations, graphing, and evaluating functions smoothly.

Combining Like Terms

Combining like terms is a crucial strategy in algebra for simplifying expressions further after applying the distributive property. Like terms are the terms in an expression that have identical variable parts. This means both the variable and its exponent must match.Looking at our simplified expression from the previous section, \(6xy + 18x\), we need to determine whether these terms can be combined.

• For terms to be combined, they must have the same variable parts. Here, \(6xy\) and \(18x\) are not like terms because \(xy\) is different from \(x\).
• This means there is no further combination possible in this particular instance.

Since no like terms are in this expression, it cannot be simplified further by combining.
Knowing how to identify and combine like terms is useful for making expressions manageable and solving equations efficiently.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations (such as addition, subtraction, multiplication, and division) combined together to represent a mathematical concept. They are the building blocks of algebra and are used extensively to describe mathematical relationships and solve problems.In the expression \(2x(3y + 9)\), there are several components:

• **Coefficient**: Numbers like \(2\) and \(9\) are coefficients, showing how many times the variable parts are used.
• **Variables**: Letters like \(x\) and \(y\) are variables, representing unknown values or quantities.
• **Terms**: Each expression is made up of terms. In the simplified expression \(6xy + 18x\), \(6xy\) and \(18x\) are terms.

Operations, such as distributing and combining like terms, help manipulate these expressions into simpler forms beneficial for solving algebraic problems.
Understanding algebraic expressions and their components is essential for tackling more complex algebraic tasks and equations.