Problem 45 - Elementary Algebra

Question

For the following problems, rewrite each phrase using algebraic notation. 3 times \((x+y)\)

Step-by-Step Solution

Answer

Question: Rewrite the phrase "3 times (x+y)" in algebraic notation. Answer: $$3x + 3y$$

Step 1: Identify the algebraic operation

In the phrase, "3 times \((x+y)\)", the operation is multiplication. We will rewrite this as a product.

Step 2: Rewrite the phrase with algebraic notation

We can rewrite "3 times \((x+y)\)" using the multiplication operation. In algebra, we simply write the coefficient (the number) next to the variable or expression in parentheses to indicate multiplication. Thus, the phrase becomes: $$3(x+y)$$

Step 3: Distribute the coefficient

Now, we can distribute the coefficient (3) to each term inside the parentheses. This means multiplying 3 by both \(x\) and \(y\) and combining the terms. It results in: $$3x + 3y$$

Step 4: Final Answer

The phrase "3 times \((x+y)\)" can be rewritten in algebraic notation as: $$3x + 3y$$

Key Concepts

Distributive Property

Understanding the distributive property is essential in algebra, particularly when dealing with expressions involving parentheses. The property tells us that if you have a number outside of parentheses (like the 3 in our problem), you can 'distribute' it to every term inside the parentheses.

In mathematical terms, we say that for all numbers a, b, and c: \[ a(b + c) = ab + ac \] When we apply this property to the exercise \(3(x + y)\), we multiply 3 by each term inside the parentheses, which gives us the expression \(3x + 3y\). This simplification is very helpful when you want to evaluate an expression at certain values or solve an equation where the terms are inside parentheses.

In short, the distributive property allows us to 'break apart' expressions to make them easier to work with.

Multiplication in Algebra

When it comes to multiplication in algebra, the process isn't all too different from what you learned in arithmetic. However, instead of just numbers, we are now dealing with variables, coefficients, and sometimes entire expressions. An important part of translating a math phrase to algebraic notation is recognizing when the multiplication operation is being used.

For instance, the phrase '3 times \(x+y\)' hints at multiplication (indicated by 'times'). Remember, in algebra, we often just place the number (known as the coefficient) next to the variable or the parentheses without using any symbol for multiplication. Knowing this helps us transition from simple numerical multiplication to working with expressions involving variables.

Rewriting Expressions

Rewriting algebraic expressions is a vital skill in solving equations and simplifying problems. When we get an algebraic phrase, like '3 times \(x+y\),' our goal is to write it in a form that is concise and ready for further mathematical operations. In algebra, expressions can often be rewritten in multiple ways, but some forms are more useful than others depending on the context.

In the given exercise, the transition from a phrase to the algebraic notation \(3(x+y)\) was the first step. After applying the distributive property, we get \(3x+3y\), which is a simplified version that clearly shows each term involved. This practice of rewriting expressions opens the door to advanced manipulation of equations and functions later in algebra.