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Chapter 2: Problem 10

Simplify each expression, if possible. $$\frac{4}{5} x(y+1)-\frac{9}{5} y(x-1)$$

Short Answer

Expert verified
The simplified form of the given expression is \(-xy+\frac{4}{5}x+\frac{9}{5}y\)

Step by step solution

01

Distribute the Coefficient

For each term, apply the distributive property of multiplication over addition which states that for all real numbers \(x\), \(y\), and \(z\), we have \(x (y + z) = xy + xz\). Hence, the given expression \(\frac{4}{5}脳(y+1)鈭抃frac{9}{5}脳(x鈭�1)\) becomes \(\frac{4}{5}xy+\frac{4}{5}x鈭抃frac{9}{5}xy+\frac{9}{5}y\)
02

Combine Like Terms

Combine the similar terms (terms having similar variables), so \(\frac{4}{5}xy - \frac{9}{5}xy\) and \(\frac{4}{5}x + \frac{9}{5}y\). So, the expression becomes \((\frac{4}{5}-\frac{9}{5})xy+\frac{4}{5}x+\frac{9}{5}y\) which simplifies to \(-xy+\frac{4}{5}x+\frac{9}{5}y\)
03

Final Simplification

On observing, it is seen that there is no more simplification could be done to this expression. So, the final answer is \(-xy+\frac{4}{5}x+\frac{9}{5}y\)

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Key Concepts

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

Distributive Property
The distributive property is a fundamental concept in algebra that allows us to multiply a single term across a sum or difference inside parentheses. In other words, when we have an expression like \(a(b + c)\), it can be rewritten as \(ab + ac\). This property is incredibly useful for breaking down and simplifying expressions.
In the given exercise, the expression \(\frac{4}{5} x(y+1)-\frac{9}{5} y(x-1)\) requires us to distribute \(\frac{4}{5}x\) across \((y+1)\) and \(-\frac{9}{5}y\) across \((x-1)\).
  • The first distribution results in \(\frac{4}{5}xy + \frac{4}{5}x\).
  • The second distribution gives us \(-\frac{9}{5}xy + \frac{9}{5}y\).
This step transforms a complicated expression into a simpler one, setting the stage for further simplification.
Combining Like Terms
Combining like terms is a process used to simplify expressions by merging terms that have identical variable parts. In our exercise, we initially have \(\frac{4}{5}xy + \frac{4}{5}x - \frac{9}{5}xy + \frac{9}{5}y\).
  • To combine like terms, look for terms with the same variable and the same exponents. Here, \(\frac{4}{5}xy\) and \(-\frac{9}{5}xy\) are like terms because they share the exact variable configuration.
By performing the arithmetic, we combine these terms: \([\frac{4}{5} - \frac{9}{5}]xy\) simplifies to \(-xy\).
Now, you're left with \(-xy + \frac{4}{5}x + \frac{9}{5}y\), which conveniently separates out the different variable components.
Simplification Process
The simplification process involves reducing an algebraic expression to its most concise form. In this exercise, after distributing and combining like terms, we are left with \(-xy + \frac{4}{5}x + \frac{9}{5}y\). This expression is already simplified, meaning there are no further operations to be performed.
Why is this the simplest form?
  • No further like terms can be combined because each term \(-xy\), \(\frac{4}{5}x\), and \(\frac{9}{5}y\) is distinct.
  • The coefficients are fully reduced, and every term stands alone with its variables.
The final goal of simplification is to make expressions easier to work with or evaluate, and to provide a clear form that often represents the most straightforward way to understand the relationship between the terms.

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