Problem 87
Question
A student said that the expression \((5 x)(3 y)\) cannot be simplified because \(5 x\) and \(3 y\) are not like terms. Explain why the student is wrong.
Step-by-Step Solution
Verified Answer
The expression \( (5x)(3y) \) simplifies to \( 15xy \).
1Step 1: Understand the problem
The student believes that the expression \( (5x)(3y) \) cannot be simplified because \(5x\) and \(3y\) are not like terms. The goal is to explain why this is incorrect.
2Step 2: Review the properties of multiplication
Multiplication of expressions involves multiplying both the coefficients and the variables. In this case, \(5x\) and \(3y\) are multiplied together.
3Step 3: Multiply the coefficients
Multiply the numerical coefficients: \(5 \times 3 = 15\).
4Step 4: Multiply the variables
Since the variables \(x\) and \(y\) are different, they are multiplied together to become \(xy\).
5Step 5: Combine the results
Combine the multiplied coefficients and variables: \(15xy\).
6Step 6: Conclusion
The expression \( (5x)(3y) \) simplifies to \(15xy\), demonstrating that the student’s initial claim was incorrect.
Key Concepts
Multiplication of TermsCoefficientsVariables
Multiplication of Terms
When simplifying algebraic expressions, one common operation is the multiplication of terms. In the given problem, the student needs to multiply \(5x\) by \(3y\). This involves two main components: coefficients and variables. Let’s break it down.
First, understand that multiplication affects all parts of the expression. This includes both the numeric (coefficient) part and the alphabetic (variable) part. Instead of looking for 'like terms,' which applies more directly when adding or subtracting expressions, focus on how each part combines under multiplication.
First, understand that multiplication affects all parts of the expression. This includes both the numeric (coefficient) part and the alphabetic (variable) part. Instead of looking for 'like terms,' which applies more directly when adding or subtracting expressions, focus on how each part combines under multiplication.
Coefficients
Coefficients are the numerical parts of the terms in an algebraic expression. Here, \(5\) in \(5x\) and \(3\) in \(3y\) are the coefficients. When multiplying terms, you multiply the coefficients first.
For example:
This is a straightforward multiplication of the numbers, following the usual arithmetic rules. It's important to do this step first because it simplifies the rest of the expression.
For example:
- Multiplying \(5\) by \(3\) gives us \(15\).
This is a straightforward multiplication of the numbers, following the usual arithmetic rules. It's important to do this step first because it simplifies the rest of the expression.
Variables
Variables are the letters that represent unknown quantities in algebraic expressions. They can be the same or different in each term. In our example, \(x\) in \(5x\) and \(y\) in \(3y\) are the variables.
After multiplying the coefficients, multiply the variables:
\(x \times y = xy\)
These variables stay together as \(xy\) because they are not 'like terms' that you combine through addition or subtraction.
Combining the product of the coefficients and the product of the variables gives you the simplified expression: \( 15xy \). This shows that even though \(5x\) and \(3y\) are not 'like terms' for addition, they can still be multiplied to produce a new term. Hence, the student's initial claim was incorrect.
After multiplying the coefficients, multiply the variables:
\(x \times y = xy\)
These variables stay together as \(xy\) because they are not 'like terms' that you combine through addition or subtraction.
Combining the product of the coefficients and the product of the variables gives you the simplified expression: \( 15xy \). This shows that even though \(5x\) and \(3y\) are not 'like terms' for addition, they can still be multiplied to produce a new term. Hence, the student's initial claim was incorrect.