Problem 67 - Algebra for College Students

Question

Simplify by removing the inner parentheses first and working outward. $$[7 x y-(2 x-3 x y+y)]-[3 x-(x-10 x y-y)]$$

Step-by-Step Solution

Answer

The simplified expression is \(-4x - 2y\).

Step 1: Simplify within the First Set of Parentheses

Start with the expression inside the first brackets: \([7xy - (2x - 3xy + y)]\). Distribute the minus sign across the inner parentheses: \(= 7xy - 2x + 3xy - y\). Combine the like terms: \(= 10xy - 2x - y\).

Step 2: Simplify within the Second Set of Parentheses

Consider the expression inside the second brackets: \([3x - (x - 10xy - y)]\). Distribute the minus sign: \(= 3x - x + 10xy + y\). Combine the like terms: \(= 2x + 10xy + y\).

Step 3: Combine and Simplify the Entire Expression

Now, substitute the simplified expressions back into the equation: \([10xy - 2x - y] - [2x + 10xy + y]\). Distribute the minus sign to the second group: \(= 10xy - 2x - y - 2x - 10xy - y\). Combine the like terms: \(= -4x - 2y\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra. It allows you to break down expressions that involve multiplication over addition or subtraction. In simple terms, it lets us multiply a single term by each term inside a parenthesis.
For example, in the expression \(a(b + c)\), the distributive property lets us expand it to \(ab + ac\).
When applying the distributive property, be attentive to signs. A common pitfall is neglecting the minus sign, which can change the entire outcome if not correctly distributed.
When you see an expression like \, \(- (2x - 3xy + y)\), it means you need to multiply \(-1\) by each term within the parentheses:

• The \(2x\) becomes \(-2x\)
• The \(3xy\) turns into \(+3xy\)
• The \(y\) becomes \(-y\)

Distribing across negative signs is like flipping the sign of each term inside!

Combining Like Terms

Combining like terms is the process of simplifying expressions by merging terms that have the same variable parts. Like terms have identical variables raised to the same power. For instance, \(3x\) and \(5x\) are like terms because they both contain \(x\) to the first power.
When combining like terms, you add or subtract the numerical coefficients while keeping the variable part the same.

• If you have \(7xy\) and \(3xy\), you can combine them to get \(10xy\).
• Similarly, \(-2x\) and \(-x\) combine to \(-3x\).

It's essential to carefully identify and pair like terms to correctly simplify the expression. Failure to do so can lead to incorrect results.
In the original problem, after distributing, carefully identify and combine like terms:
\(7xy - 2x + 3xy - y \) simplifies to \(10xy - 2x - y\) by combining \(7xy\) with \(3xy\).

Algebraic Expressions

An algebraic expression includes numbers, variables, and operation symbols, such as addition or subtraction, but doesn't have an equals sign. These expressions form the backbone of algebra and are used to describe relationships or situations numerically.

• Examples of simple algebraic expressions include: \(2x + 3\), \(4xy - 7\), or \(xy - z + 2\).

There are no restrictions on the number of terms that an algebraic expression can include. They can be as simple or as complex as needed depending on the scenario.
In more complex expressions, like the one in the original exercise, you often need to first simplify each part by handling operations such as distributing and combining like terms.
The primary aim when working with algebraic expressions is simplification. This is done by arranging and reducing the terms to their simplest form, as guided by various algebraic rules and properties. Simplified expressions are easier to interpret and solve in algebraic equations.