Problem 16
Question
A student simplified \(3 x+7 y\) as \(10 x y\). Explain why this is not correct.
Step-by-Step Solution
Verified Answer
The expression \(3x + 7y\) cannot be simplified to \(10xy\) because addition of unlike terms cannot be turned into a product.
1Step 1: Identify the Original Expression
The original expression given is: \(3x + 7y\)
2Step 2: Understand the Simplification
The student simplified \(3x + 7y\) as \(10xy\).
3Step 3: Analyze the Terms
Break down the terms in the original expression. \(3x\) and \(7y\) are individual terms that involve addition of products of variables with coefficients.
4Step 4: Address the Incorrect Step
The mistake lies in combining the two terms into a single product. Addition in algebra does not allow combining unlike terms (terms with different variables) through multiplication. Hence, \(3x + 7y\) cannot be simplified to \(10xy\).
5Step 5: Proper Simplification
The expression \(3x + 7y\) is already in its simplest form since the terms \(3x\) and \(7y\) cannot be further simplified or combined.
Key Concepts
Combining Like TermsAddition in AlgebraAlgebraic Simplification
Combining Like Terms
In algebra, combining like terms is essential for simplifying expressions. Like terms are terms that have exactly the same variable factors. For example, in the expression \(3x + 7y\), \(3x\) and \(7y\) are not like terms because they have different variables. Only terms with the same variable(s) and exponents can be combined.
For instance, in the expression \(4x + 3x\), the terms can be combined because both have the variable \(x\). Combining these would give \(7x\).
Always look for and combine only those terms with the same variable. This keeps the expression as simplified as possible by reducing the number of terms.
For instance, in the expression \(4x + 3x\), the terms can be combined because both have the variable \(x\). Combining these would give \(7x\).
Always look for and combine only those terms with the same variable. This keeps the expression as simplified as possible by reducing the number of terms.
Addition in Algebra
Addition in algebra works similarly to regular addition, but you can only add like terms. When adding, each term's coefficient is added to the coefficients of like terms. For the expression \(3x + 7y\), these terms involve addition but cannot be combined further.
Adding \(3x\) with any other \(x\)-term (e.g., \(4x\)) results in \(7x\), where the exponents and variables are the same.
Remember that unlike terms (such as \(x\)-terms and \(y\)-terms) cannot be added. Keeping track of terms ensures you maintain the expression accurately without creating incorrect terms like \(10xy\).
Adding \(3x\) with any other \(x\)-term (e.g., \(4x\)) results in \(7x\), where the exponents and variables are the same.
Remember that unlike terms (such as \(x\)-terms and \(y\)-terms) cannot be added. Keeping track of terms ensures you maintain the expression accurately without creating incorrect terms like \(10xy\).
Algebraic Simplification
Algebraic simplification involves reducing expressions to their simplest forms. For the expression \(3x + 7y\), both the terms \(3x\) and \(7y\) are already as simple as possible.
Simplification means:
Before combining terms, always check if they are like terms. This ensures the simplification is done correctly and efficiently.
Simplification means:
- Combining like terms
- Removing any unnecessary parentheses
- Reducing coefficients and constants if possible
Before combining terms, always check if they are like terms. This ensures the simplification is done correctly and efficiently.