Problem 73 - Elementary Algebra

Question

For the following problems, use the distributive property to expand the expressions. $$ 3 y(2 x+4 z+5 w) $$

Step-by-Step Solution

Answer

Question: Expand the given expression using the distributive property: \(3y(2x+4z+5w)\). Answer: \(6xy + 12yz + 15yw\).

Step 1: Distribute 3y to 2x

Multiply 3y by 2x to get 6xy.

Step 2: Distribute 3y to 4z

Multiply 3y by 4z to get 12yz.

Step 3: Distribute 3y to 5w

Multiply 3y by 5w to get 15yw.

Step 4: Combine terms

Combine the resulting terms to get the expanded expression: \(6xy + 12yz + 15yw\).

Key Concepts

Algebraic Expressions

Algebraic expressions are like math sentences that combine numbers and letters (or variables) with operations like addition, subtraction, multiplication, and division. They allow us to represent real-world situations in a mathematical form. For example, in the expression \(3y(2x + 4z + 5w)\), \(y\), \(x\), \(z\), and \(w\) are variables that can take on different values. The numbers like \(3\), \(2\), \(4\), and \(5\) are coefficients that multiply the variables. Understanding algebraic expressions helps you solve equations and model scenarios using math.

Think of algebraic expressions as recipes where numbers and letters are ingredients, and the operations are the steps. Each term in an expression can be likened to an individual dish in a meal. It's essential to understand that an algebraic expression isn't an equation since it doesn't equate to anything specific unless stated otherwise.

Multiplication in Algebra

Multiplication in algebra is slightly different from regular multiplication since it involves variables as well as numbers. The operation follows the same rules as arithmetic but introduces new concepts like the distributive property. This property allows us to multiply a single term by a grouping of terms inside parentheses. In our expression \(3y(2x + 4z + 5w)\), "distributing" means that we multiply \(3y\) by each term within the parentheses, one at a time.

To break it down:

• Multiply \(3y\) by \(2x\) to get \(6xy\).
• Multiply \(3y\) by \(4z\) to get \(12yz\).
• Multiply \(3y\) by \(5w\) to get \(15yw\).

Multiplying variables along with numbers means combining coefficients (numbers) and listing the variables alphabetically in the product, as seen above. It simplifies understanding complex algebraic expressions.

Expanding Expressions

Expanding expressions involves breaking down an expression into simpler terms, typically to make it easier to work with. This process often uses the distributive property and aims to "spread out" or "expand" an expression that includes parentheses, like our given expression \(3y(2x + 4z + 5w)\).

An expanded expression has each term individually laid out without any parentheses. The steps include multiplying each term inside the parentheses by the factor outside, then writing the resulting terms side by side separated by plus signs if they are positive, or minus signs if they are negative.

• Starting with the multiplication of terms, like \(6xy\), \(12yz\), and \(15yw\).
• Each term comes from multiplying \(3y\) with each component inside the parentheses.
• The final expanded expression is a combination of these individual terms forming \(6xy + 12yz + 15yw\).

This method clarifies the algebraic expression and prepares it for further operations like addition, subtraction, or even factoring in future algebra problems.