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Chapter 13: Problem 39

Find each difference. $$(4 x+y)-(5 x+y)$$

Short Answer

Expert verified
The difference is \(-x\).

Step by step solution

01

Distribute the Negative Sign

Start by applying the minus sign to both terms inside the parentheses: \[(4x + y) - (5x + y) = (4x + y) - 5x - y \]
02

Combine Like Terms

Now, group the similar terms together:\[(4x - 5x) + (y - y) \]
03

Simplify the Expressions

Simplify each group of like terms:- For the terms involving \(x\): \(4x - 5x = -x\)- For the terms involving \(y\): \(y - y = 0\)Thus, the result will be:\[-x + 0 = -x\]
04

Final Simplification

The expression simplifies to just:\[-x\] since adding zero will not change the value of \(-x\).

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Key Concepts

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

The Distributive Property
In algebra, the distributive property is a fundamental tool for simplifying expressions. It involves distributing a single term across the terms inside a set of parentheses. This process is used when you need to multiply or subtract terms that are grouped together.
For instance, in the expression \((4x + y) - (5x + y)\), the distributive property is applied by the negative sign to the second set of terms.
This is because you're essentially multiplying \(-1\) by each term inside the parentheses:
  • The \(-1\) times the \(5x\) becomes \(-5x\).
  • The \(-1\) times the \(y\) becomes \(-y\).
After applying the distributive property, the expression changes. Breaking it down helps in organizing the terms properly for the next steps.
Combining Like Terms
Once the distributive property has been applied, it's time to combine like terms. Like terms are terms that have the same variable raised to the same power. For our example, that means:
  • Terms involving \(x\) are \(4x\) and \(-5x\).
  • Terms involving \(y\) are \(y\) and \(-y\).
By grouping these like terms, you effectively simplify your expression. In our exercise, \(4x\) and \(-5x\) combine to give \(-x\). Similarly, the \(y\) terms cancel each other out since \(y - y = 0\).
Combining like terms makes solving expressions quicker and ensures that you keep track of all similar mathematical entities together in one go.
Simplifying Expressions
The culmination of using the distributive property and combining like terms results in a simplified expression. Simplification means reducing your expression to its simplest form, without changing its value.
For the exercise at hand, once the terms \(x\) and \(y\) have been combined, you're left with \(-x + 0\).
In math, adding zero to any term leaves the term unchanged, so \(-x + 0\) simply becomes \(-x\). This is the result of the simplification process.
Remember:
  • Simplifying expressions helps in solving equations faster.
  • It clarifies the mathematical problem by reducing complexities.
The key takeaway is to learn the process step-by-step and apply these methods universally to algebraic problems.

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