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Chapter 13: Problem 18

Find each difference. $$(2 x+3 y)-(x-y)$$

Short Answer

Expert verified
The difference is \(x + 4y\).

Step by step solution

01

Distribute the negative sign

When subtracting an expression, distribute the negative sign across the terms in the subtracted expression. In this case: \((2x + 3y) - (x - y) = 2x + 3y - x + y\).
02

Combine like terms

Combine the terms with the same variables. The expression \(2x + 3y - x + y\) can be simplified further:- Combine the \(x\) terms: \(2x - x = x\).- Combine the \(y\) terms: \(3y + y = 4y\).Thus, the simplified expression is \(x + 4y\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Expressions
Simplifying expressions means making them easier to understand and work with by performing operations and reducing terms. In mathematical problems, simplifying an expression can often make the problem easier to solve or comprehend.
  • Start by removing any unnecessary parts of the expression.
  • Look out for terms that can be simplified using arithmetic operations, such as addition and subtraction.
  • Use mathematical properties like the distributive property to transform expressions.
Let's consider our example: with the expression \( (2x + 3y) - (x - y) \), the goal was to simplify it for easier handling. The term \( (x - y) \) was initially subtracted from \( 2x + 3y \), which might not be immediately clear.
By distributing the negative sign into the second parentheses, it transforms into \( 2x + 3y - x + y \).
This change aids in more straightforward computation as the terms can now be easily grouped.
Combining Like Terms
Combining like terms involves merging terms within an expression that share the same variables raised to the same power. This step is crucial to simplifying algebraic expressions.
  • Terms must be "like", meaning their variable components match exactly.
  • Add or subtract the coefficients of the like terms while maintaining their variable parts.
In the expression \( 2x + 3y - x + y \), it includes like terms that can be combined to further simplify.
The \( x \) terms \( 2x \) and \(-x \) can be combined to equal \( x \).
Similarly, the \( y \) terms \( 3y \) and \( y \) combine to make \( 4y \).
After combining these like terms, we achieve the final simplified form: \( x + 4y \).
This final expression is not only simpler, but also neatly grouped into a more manageable form.
Distributive Property
The distributive property is a handy mathematical rule that allows us to simplify expressions involving multiplication over addition or subtraction. This rule states: \( a(b + c) = ab + ac \) or \( a(b - c) = ab - ac \).
  • It's especially useful when dealing with parentheses and multiple terms.
  • Distribute the multiplying factor across each term inside the parentheses.
In our original exercise, the step where we distribute the negative sign illustrates the distributive property:
The expression \((2x + 3y) - (x - y)\) was transformed by distributing the negative sign as \( 2x + 3y - x + y \).
This step helps remove the parentheses, making it easier to identify like terms to combine.
This layer of simplification establishes a clear pathway to tidier calculations.

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