Problem 68 - Algebra for College Students

Question

Simplify by removing the inner parentheses first and working outward. $$[9 x y-(4 x+x y-y)]-[4 y-(2 x-x y+6 y)]$$

Step-by-Step Solution

Answer

The simplified expression is \(8xy - 6x + 11y\).

Step 1: Address the first set of inner parentheses

Look at the expression inside the first inner parentheses: \(9xy - (4x + xy - y)\). Distribute the negative sign to each term inside these parentheses: \(9xy - 4x - xy + y\).

Step 2: Address the second set of inner parentheses

Now look inside the second inner parentheses: \( -[4y - (2x - xy + 6y)] \). Distribute the negative sign inside these parenthesis first: \(-4y + 2x - xy - 6y\).

Step 3: Remove the outer brackets

Combine the results from Steps 1 and 2 to remove the outer brackets: \([9xy - 4x - xy + y] - [2x - xy - 10y]\).This becomes: \(9xy - 4x - xy + y - 2x + xy + 10y\).

Step 4: Combine like terms

Now that the expression is simplified from parentheses, combine the like terms:- For \(xy\), we have \(9xy\) and \(-xy\) which simplifies to \(8xy\).- For \(x\), we have \(-4x\) and \(-2x\) to become \(-6x\).- For \(y\), we have \(y + 10y\) to simplify to \(11y\).

Step 5: Write the final simplified expression

The expression \(8xy - 6x + 11y\) is the simplified form of the original expression by removing all parentheses and combining like terms.

Key Concepts

Simplifying Expressions

Simplifying algebraic expressions is all about making them easier to read and understand. At first glance, an algebraic expression might look pretty complicated with several terms, brackets, and even various operations. But the good news is that with simplification, we can tidy it all up! When simplifying expressions, we follow systematic steps to transform the expression into a simpler form that represents the same quantity.

To simplify an expression:

• Look out for unnecessary parentheses.
• Apply operations such as distributing negatives or factors as necessary.
• Combine similar terms to further reduce the expression.

By doing this, the expression becomes clearer and more straightforward, just like decluttering a messy room to make everything more accessible and organized.

Distributive Property

The distributive property is a handy tool in algebra that allows us to simplify expressions involving parentheses. It states that multiplying a number or term by a group of terms inside parentheses is the same as individually multiplying each term inside the parentheses by that number or term.

Usually, this is expressed as:

• For a single term: \( a(b + c) = ab + ac \)
• For subtraction: \( a(b - c) = ab - ac \)

In our original problem, instead of a multiplying number, we dealt with distributing a negative sign, which functions similarly. For example, applying the property to \(- (4x + xy - y)\) results in \(-4x - xy + y\). Likewise, in the second bracket, it changes \(- (2x - xy + 6y)\) to \(2x - xy - 6y\).

This process helps dissolve brackets, making it easier to work with the rest of the expression without unnecessary layers.

Combining Like Terms

After using the distributive property, the next step to simplify the expression is to combine like terms. Like terms in an expression share the same variable factors. This means they have identical variable parts, although coefficients (the number in front of the variables) can differ.

Here's how to do it:

• Identify terms that have the same variable.
• Add or subtract their coefficients according to the operations present.

In our example, we found like terms for each variable:

- For the terms involving \(xy\), we had \(9xy\) and \(-xy\), which combined to form \(8xy\).
- For the \(x\) terms, \(-4x\) and \(-2x\) add up to \(-6x\).
- The \(y\) terms, \(y\) and \(10y\), chimed together into \(11y\).

By identifying and combining like terms, expressions shrink further, making the final simplified version easy to understand and work with.